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Pythagorean theorem

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Relation between sides of a right triangle

Pythagorean theorem
TypeTheorem
FieldEuclidean geometry
StatementThe sum of the areas of the two squares on the legs (a andb) equals the area of the square on the hypotenuse (c).
Symbolic statementa2 +b2 =c2
Generalizations
Consequences
Geometry
Stereographic projection from the top of a sphere onto a plane beneath it
Geometers

Inmathematics, thePythagorean theorem orPythagoras's theorem is a fundamental relation inEuclidean geometry between the three sides of aright triangle. It states that the area of thesquare whose side is thehypotenuse (the side opposite theright angle) is equal to the sum of the areas of the squares on the other two sides.[1]

Thetheorem can be written as anequation relating the lengths of the sidesa,b and the hypotenusec, sometimes called thePythagorean equation:[2]a2+b2=c2.{\displaystyle a^{2}+b^{2}=c^{2}.}The theorem is named for theGreek philosopherPythagoras, born around 570 BC. The theorem has beenproved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including bothgeometric proofs andalgebraic proofs, with some dating back thousands of years.

WhenEuclidean space is represented by aCartesian coordinate system inanalytic geometry,Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.

The theorem can begeneralized in various ways: tohigher-dimensional spaces, tospaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all butn-dimensional solids.

Proofs using constructed squares

Rearrangement proof of the Pythagorean theorem.
(The area of the white space remains constant throughout the translation rearrangement of the triangles. At all moments in time, the area is alwaysc2. And likewise, at all moments in time, the area is alwaysa2 +b2.)

Rearrangement proofs

In one rearrangement proof, two squares are used whose sides have a measure ofa+b{\displaystyle a+b}and which contain four right triangles whose sides area,b andc, with the hypotenuse beingc. In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are lengthc. Each outer square has an area of(a +b)2 as well as2ab +c2, with2ab representing the total area of the four triangles. Within the big square on the left side, the four triangles are moved to form two similar rectangles with sides of lengtha andb. These rectangles in their new position have now delineated two new squares, one having side lengtha is formed in the bottom-left corner, and another square of side lengthb formed in the top-right corner. In this new position, this left side now has a square of area(a +b)2 as well as2ab +a2 +b2. Since both squares have the area of(a +b)2 it follows that the other measure of the square area also equal each other such that2ab +c2 =ab +a2 +b2. With the area of the four triangles removed from both side of the equation what remains isa2 +b2 =c2.[3]

In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square. In this way they also form two boxes, this time in consecutive corners, with areasa2 andb2 which will again lead to a second square of with the area2ab +a2 +b2.

English mathematicianSir Thomas Heath gives this proof in his commentary on Proposition I.47 inEuclid'sElements, and mentions the proposals of German mathematiciansCarl Anton Bretschneider andHermann Hankel that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean proof, but acknowledges from the outset of his discussion "that the Greek literature which we possess belonging to the first five centuries after Pythagoras contains no statement specifying this or any other particular great geometric discovery to him."[4] Recent scholarship has cast increasing doubt on any sort of role for Pythagoras as a creator of mathematics, although debate about this continues.[5]

Algebraic proofs

Diagram of the two algebraic proofs

The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with sidec, as shown in the lower part of the diagram.[6] This results in a larger square, with sidea +b and area(a +b)2. The four triangles and the square sidec must have the same area as the larger square,(b+a)2=c2+4ab2=c2+2ab,{\displaystyle (b+a)^{2}=c^{2}+4{\frac {ab}{2}}=c^{2}+2ab,}givingc2=(b+a)2−2ab=b2+2ab+a2−2ab=a2+b2.{\displaystyle c^{2}=(b+a)^{2}-2ab=b^{2}+2ab+a^{2}-2ab=a^{2}+b^{2}.}

A similar proof uses four copies of a right triangle with sidesa,b andc, arranged inside a square with sidec as in the top half of the diagram.[7] The triangles are similar with area⁠1/2⁠ab, while the small square has sideb −a and area(b −a)2. The area of the large square is therefore(b−a)2+4ab2=(b−a)2+2ab=b2−2ab+a2+2ab=a2+b2.{\displaystyle (b-a)^{2}+4{\frac {ab}{2}}=(b-a)^{2}+2ab=b^{2}-2ab+a^{2}+2ab=a^{2}+b^{2}.}

But this is a square with sidec and areac2, soc2=a2+b2.{\displaystyle c^{2}=a^{2}+b^{2}.}

Other proofs of the theorem

This theorem may have more known proofs than any other (thelaw ofquadratic reciprocity being another contender for that distinction); the bookThe Pythagorean Proposition contains 370 proofs.[8]

Proof using similar triangles

In this section, and as usual in geometry, a "word" of two capital letters, such asAB denotes the length of theline segment defined by the points labeled with the letters, and not a multiplication. So,AB2 denotes the square of the lengthAB and not the productA ×B2.
Proof using similar triangles

This proof is based on theproportionality of the sides of threesimilar triangles, that is, upon the fact that theratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles.

LetABC represent a right triangle, with theright angle located atC, as shown on the figure. Draw thealtitude from pointC, and callH its intersection with the sideAB. PointH divides the length of the hypotenusec into partsd ande. The new triangle,ACH, issimilar to triangleABC, because they both have a right angle (by definition of the altitude), and they share the angle atA, meaning that the third angle will be the same in both triangles as well, marked asθ in the figure. By a similar reasoning, the triangleCBH is also similar toABC. The proof of similarity of the triangles requires thetriangle postulate: The sum of the angles in a triangle is two right angles, and is equivalent to theparallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides:BCAB=BHBCACAB=AHAC.{\displaystyle {\begin{aligned}{\frac {BC}{AB}}&={\frac {BH}{BC}}\\{\frac {AC}{AB}}&={\frac {AH}{AC}}.\end{aligned}}}

The first result equates thecosines of the anglesθ, whereas the second result equates theirsines.

These ratios can be written asBC2=AB×BH,AC2=AB×AH.{\displaystyle {\begin{aligned}BC^{2}&=AB\times BH,\\AC^{2}&=AB\times AH.\end{aligned}}}Summing these two equalities results inBC2+AC2=AB×BH+AB×AH=AB(AH+BH),{\displaystyle {\begin{aligned}BC^{2}+AC^{2}&=AB\times BH+AB\times AH\\&=AB(AH+BH),\end{aligned}}}which, after simplification, demonstrates the Pythagorean theorem:BC2+AC2=AB2.{\displaystyle BC^{2}+AC^{2}=AB^{2}.}

The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. Oneconjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in theElements, and that the theory of proportions needed further development at that time.[9]

Einstein's proof by dissection without rearrangement

Right triangle on the hypotenuse dissected into two similar right triangles on the legs, according to Einstein's proof.

Albert Einstein gave a proof by dissection in which the pieces do not need to be moved.[10] Instead of using a square on the hypotenuse and two squares on the legs, one can use any other shape that includes the hypotenuse, and twosimilar shapes that each include one of two legs instead of the hypotenuse (seeSimilar figures on the three sides). In Einstein's proof, the shape that includes the hypotenuse is the right triangle itself. The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts. Those two parts have the same shape as the original right triangle,and have the legs of the original triangle as their hypotenuses, and the sum of their areas is that of the original triangle. Because the ratio of the area of a right triangle to the square of its hypotenuse is the same for similar triangles, the relationship between the areas of the three triangles holds for the squares of the sides of the large triangle as well.

Euclid's proof

Proof in Euclid'sElements

In outline, here is how the proof inEuclid'sElements proceeds. The large square is divided into a left and right rectangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side. These two triangles are shown to becongruent, proving this square has the same area as the left rectangle. This argument is followed by a similar version for the right rectangle and the remaining square. Putting the two rectangles together to reform the square on the hypotenuse, its area is the same as the sum of the area of the other two squares. The details follow.

LetA,B,C be thevertices of a right triangle, with a right angle atA. Drop a perpendicular fromA to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs.

For the formal proof, we require four elementarylemmata:

  1. If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent (side-angle-side).
  2. The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
  3. The area of a rectangle is equal to the product of two adjacent sides.
  4. The area of a square is equal to the product of two of its sides (follows from 3).

Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square.[11]

Illustration including the new lines
Showing the two congruent triangles of half the area of rectangleBDLK and squareBAGF

The proof is as follows:

  1. LetACB be a right-angled triangle with right angleCAB.
  2. On each of the sidesBC,AB, andCA, squares are drawn,CBDE,BAGF, andACIH, in that order. The construction of squares requires the immediately preceding theorems in Euclid, and depends upon the parallel postulate.[12]
  3. FromA, draw a line parallel toBD andCE. It will perpendicularly intersectBC andDE atK andL, respectively.
  4. JoinCF andAD, to form the trianglesBCF andBDA.
  5. AnglesCAB andBAG are both right angles; thereforeC,A, andG arecollinear.
  6. AnglesCBD andFBA are both right angles; therefore angleABD equals angleFBC, since both are the sum of a right angle and angleABC.
  7. SinceAB is equal toFB,BD is equal toBC and angleABD equals angleFBC, triangleABD must be congruent to triangleFBC.
  8. SinceA-K-L is a straight line, parallel toBD, then rectangleBDLK has twice the area of triangleABD because they share the baseBD and have the same altitudeBK, i.e., a line normal to their common base, connecting the parallel linesBD andAL. (lemma 2)
  9. SinceC is collinear withA andG, and this line is parallel toFB, then squareBAGF must be twice in area to triangleFBC.
  10. Therefore, rectangleBDLK must have the same area as squareBAGF =AB2.
  11. By applying steps 3 to 10 to the other side of the figure, it can be similarly shown that rectangleCKLE must have the same area as squareACIH =AC2.
  12. Adding these two results,AB2 +AC2 =BD ×BK +KL ×KC
  13. SinceBD =KL,BD ×BK +KL ×KC =BD(BK +KC) =BD ×BC
  14. Therefore,AB2 +AC2 =BC2, sinceCBDE is a square(BD ×BC =BC2).

This proof, which appears in Euclid'sElements as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares.[13][14]This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used.[15][a]

Proofs by dissection and rearrangement

Another by rearrangement is given by the middle animation. A large square is formed with areac2, from four identical right triangles with sidesa,b andc, fitted around a small central square. Then two rectangles are formed with sidesa andb by moving the triangles. Combining the smaller square with these rectangles produces two squares of areasa2 andb2, which must have the same area as the initial large square.[16]

The third, rightmost image also gives a proof. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse – or conversely the large square can be divided as shown into pieces that fill the other two. This way of cutting one figure into pieces and rearranging them to get another figure is calleddissection. This shows the area of the large square equals that of the two smaller ones.[17]

Animation showing proof by rearrangement of four identical right triangles
Animation showing another proof by rearrangement
Proof using an elaborate rearrangement

Proof by area-preserving shearing

Visual proof of the Pythagorean theorem by area-preserving shearing

As shown in the accompanying animation, area-preservingshear mappings and translations can transform the squares on the sides adjacent to the right-angle onto the square on the hypotenuse, together covering it exactly.[18] Each shear leaves the base and height unchanged, thus leaving the area unchanged too. The translations also leave the area unchanged, as they do not alter the shapes at all. Each square is first sheared into a parallelogram, and then into a rectangle which can be translated onto one section of the square on the hypotenuse.

Other algebraic proofs

A relatedproof by U.S. president James A. Garfield was published before he was elected president; while he was aU.S. representative.[19][20][21] Instead of a square it uses atrapezoid, which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in the diagram. Thearea of the trapezoid can be calculated to be half the area of the square, that is12(b+a)2.{\displaystyle {\tfrac {1}{2}}(b+a)^{2}.}

The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of⁠12{\displaystyle {\tfrac {1}{2}}}⁠, which is removed by multiplying by two to give the result.

Proof using differentials

One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employingcalculus.[22][23][24]

The triangleABC is a right triangle, as shown in the upper part of the diagram, withBC the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of lengthy, the sideAC of lengthx and the sideAB of lengtha, as seen in the lower diagram part.

Diagram for differential proof

Ifx is increased by a small amountdx by extending the sideAC slightly toD, theny also increases bydy. These form two sides of a triangle,CDE, which (withE chosen soCE is perpendicular to the hypotenuse) is a right triangle approximately similar toABC. Therefore, the ratios of their sides must be the same, that is:dydx=xy.{\displaystyle {\frac {dy}{dx}}={\frac {x}{y}}.}This can be rewritten asydy =xdx, which is adifferential equation that can be solved by direct integration:∫ydy=∫xdx,{\displaystyle \int y\,dy=\int x\,dx,}givingy2=x2+C.{\displaystyle y^{2}=x^{2}+C.}The constant can be deduced fromx = 0,y =a to give the equationy2=x2+a2.{\displaystyle y^{2}=x^{2}+a^{2}.}This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place ofdx anddy.

Converse

Theconverse of the theorem is also true:[25]

Given a triangle with sides of lengtha,b, andc, ifa2 +b2 =c2, then the angle between sidesa andb is aright angle.

For any three positivereal numbersa,b, andc such thata2 +b2 =c2, there exists a triangle with sidesa,b andc as a consequence of theconverse of the triangle inequality.

This converse appears in Euclid'sElements (Book I, Proposition 48): "If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right."[26]

It can be proved using thelaw of cosines or as follows:

LetABC be a triangle with side lengthsa,b, andc, witha2 +b2 =c2. Construct a second triangle with sides of lengtha andb containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length⁠c=a2+b2{\displaystyle \textstyle c={\sqrt {a^{2}+b^{2}}}}⁠, the same as the hypotenuse of the first triangle. Since both triangles' sides are the same lengthsa,b andc, the triangles arecongruent and must have the same angles. Therefore, the angle between the side of lengthsa andb in the original triangle is a right angle.

The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proved without assuming the Pythagorean theorem.[27][28]

Acorollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Letc be chosen to be the longest of the three sides anda +b >c (otherwise there is no triangle according to thetriangle inequality). The following statements apply:[29]

Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language:sgn⁡(α+β−γ)=sgn⁡(a2+b2−c2),{\displaystyle \operatorname {sgn} (\alpha +\beta -\gamma )=\operatorname {sgn} (a^{2}+b^{2}-c^{2}),}whereα is the angle opposite to sidea,β is the angle opposite to sideb,γ is the angle opposite to sidec, and sgn is thesign function.[30]

Consequences and uses of the theorem

Pythagorean triples

Main article:Pythagorean triple
See also:Formulas for generating Pythagorean triples

A Pythagorean triple has three positive integersa,b, andc, such thata2 +b2 =c2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.[2] Such a triple is commonly written(a,b,c). Some well-known examples are(3, 4, 5) and(5, 12, 13).

A primitive Pythagorean triple is one in whicha,b andc arecoprime (thegreatest common divisor ofa,b, andc is 1).

The following is a list of primitive Pythagorean triples with values less than 100:

(3, 4, 5),(5, 12, 13),(7, 24, 25),(8, 15, 17),(9, 40, 41),(11, 60, 61),(12, 35, 37),(13, 84, 85),(16, 63, 65),(20, 21, 29),(28, 45, 53),(33, 56, 65),(36, 77, 85),(39, 80, 89),(48, 55, 73),(65, 72, 97)

There are manyformulas for generating Pythagorean triples. Of these,Euclid's formula is the most well-known: given arbitrary positive integersm andn, the formula states that the integersa=m2−n2,b=2mn,c=m2+n2{\displaystyle a=m^{2}-n^{2},\quad b=2mn,\quad c=m^{2}+n^{2}} forms a Pythagorean triple.

Inverse Pythagorean theorem

Given aright triangle with sidesa,b,c{\displaystyle a,b,c} andaltituded (a line from the right angle and perpendicular to thehypotenusec). The Pythagorean theorem has,a2+b2=c2{\displaystyle a^{2}+b^{2}=c^{2}}while theinverse Pythagorean theorem relates the twolegsa, b to the altituded,[31]1a2+1b2=1d2{\displaystyle {\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}={\frac {1}{d^{2}}}}The equation can be transformed to,1(xz)2+1(yz)2=1(xy)2{\displaystyle {\frac {1}{(xz)^{2}}}+{\frac {1}{(yz)^{2}}}={\frac {1}{(xy)^{2}}}}wherex2 +y2 =z2 for any non-zerorealx, y , z. If thea, b, d are to beintegers, the smallest solutiona >b >d is then1202+1152=1122{\displaystyle {\frac {1}{20^{2}}}+{\frac {1}{15^{2}}}={\frac {1}{12^{2}}}}using the smallest Pythagorean triple3, 4, 5. The reciprocal Pythagorean theorem is a special case of theoptic equation1p+1q=1r{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}={\frac {1}{r}}}where the denominators are squares and also for aheptagonal triangle whose sidesp, q, r are square numbers.

Incommensurable lengths

Thespiral of Theodorus: A construction for line segments with lengths whose ratios are the square root of a positive integer

One of the consequences of the Pythagorean theorem is that line segments whose lengths areincommensurable (so the ratio of which is not arational number) can be constructed using astraightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by thesquare root operation.

The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer.[32] Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is not a perfect square, it is a realization of a length incommensurable with the unit, such as⁠2{\displaystyle {\sqrt {2}}}⁠,⁠3{\displaystyle {\sqrt {3}}}⁠,⁠5{\displaystyle {\sqrt {5}}}⁠. For more detail, seeQuadratic irrational.

Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers. The Pythagorean school dealt with proportions by comparison of integer multiples of a common subunit.[33] According to one legend,Hippasus of Metapontum (c. 470 BC) was drowned at sea for making known the existence of the irrational or incommensurable.[34]Kurt von Fritz wrote a careful discussion of Hippasus's contributions.[35]

Complex numbers

The absolute value of a complex numberz is the distancer fromz to the origin.

For anycomplex numberz=x+iy,{\displaystyle z=x+iy,}theabsolute value or modulus is given byr=|z|=x2+y2.{\displaystyle r=|z|={\textstyle {\sqrt {x^{2}+y^{2}}}}.}So the three quantities,r,x andy are related by the Pythagorean equation,r2=x2+y2.{\displaystyle r^{2}=x^{2}+y^{2}.}Note thatr is defined to be a positive number or zero butx andy can be negative as well as positive. Geometricallyr is the distance of thez from zero or the originO in thecomplex plane.

This can be generalised to find the distance between two points,z1 andz2 say. The required distance is given by|z1−z2|=(x1−x2)2+(y1−y2)2,{\displaystyle |z_{1}-z_{2}|={\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}},}so again they are related by a version of the Pythagorean equation,|z1−z2|2=(x1−x2)2+(y1−y2)2.{\displaystyle |z_{1}-z_{2}|^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}.}

Euclidean distance

Main article:Euclidean distance

The distance formula inCartesian coordinates is derived from the Pythagorean theorem.[36] If(x1,y1) and(x2,y2) are points in the plane, then the distance between them, also called theEuclidean distance, is given by

(x1−x2)2+(y1−y2)2.{\displaystyle {\sqrt {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}.}

More generally, inEuclideann-space, the Euclidean distance between two points,A=(a1,a2,…,an){\displaystyle A=(a_{1},\,a_{2},\,\dots ,\,a_{n})} andB=(b1,b2,…,bn){\displaystyle B=(b_{1},\,b_{2},\,\dots ,\,b_{n})}, is defined, by generalization of the Pythagorean theorem, as:

(a1−b1)2+(a2−b2)2+⋯+(an−bn)2=∑i=1n(ai−bi)2.{\displaystyle {\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots +(a_{n}-b_{n})^{2}}}={\sqrt {\sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}.}

If instead of Euclidean distance, the square of this value (thesquared Euclidean distance, or SED) is used, the resulting equation avoids square roots and is simply a sum of the SED of the coordinates:

(a1−b1)2+(a2−b2)2+⋯+(an−bn)2=∑i=1n(ai−bi)2.{\displaystyle (a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots +(a_{n}-b_{n})^{2}=\sum _{i=1}^{n}(a_{i}-b_{i})^{2}.}

The squared form is a smooth,convex function of both points, and is widely used inoptimization theory andstatistics, forming the basis ofleast squares.

Euclidean distance in other coordinate systems

If Cartesian coordinates are not used, for example, ifpolar coordinates are used in two dimensions or, in more general terms, ifcurvilinear coordinates are used, the formulas expressing the Euclidean distance are more complicated than the Pythagorean theorem, but can be derived from it. A typical example where the straight-line distance between two points is converted to curvilinear coordinates can be found in theapplications of Legendre polynomials in physics. The formulas can be discovered by using Pythagoras's theorem with the equations relating the curvilinear coordinates to Cartesian coordinates. For example, the polar coordinates(r,θ) can be introduced as:

x=rcos⁡θ,y=rsin⁡θ.{\displaystyle x=r\cos \theta ,\quad y=r\sin \theta .}

Then two points with locations(r1,θ1) and(r2,θ2) are separated by a distances:

s2=(x1−x2)2+(y1−y2)2=(r1cos⁡θ1−r2cos⁡θ2)2+(r1sin⁡θ1−r2sin⁡θ2)2.{\displaystyle s^{2}=(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}=(r_{1}\cos \theta _{1}-r_{2}\cos \theta _{2})^{2}+(r_{1}\sin \theta _{1}-r_{2}\sin \theta _{2})^{2}.}Performing the squares and combining terms, the Pythagorean formula for distance in Cartesian coordinates produces the separation in polar coordinates as:s2=r12+r22−2r1r2(cos⁡θ1cos⁡θ2+sin⁡θ1sin⁡θ2)=r12+r22−2r1r2cos⁡(θ1−θ2)=r12+r22−2r1r2cos⁡Δθ,{\displaystyle {\begin{aligned}s^{2}&=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\left(\cos \theta _{1}\cos \theta _{2}+\sin \theta _{1}\sin \theta _{2}\right)\\&=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos \left(\theta _{1}-\theta _{2}\right)\\&=r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos \Delta \theta ,\end{aligned}}}using the trigonometricproduct-to-sum formulas. This formula is thelaw of cosines, sometimes called the generalized Pythagorean theorem.[37] From this result, for the case where the radii to the two locations are at right angles, the enclosed angleΔθ =π/2, and the form corresponding to Pythagoras's theorem is regained:s2=r12+r22.{\displaystyle s^{2}=r_{1}^{2}+r_{2}^{2}.} The Pythagorean theorem, valid for right triangles, therefore is a special case of the more general law of cosines, valid for arbitrary triangles.

Pythagorean trigonometric identity

Main article:Pythagorean trigonometric identity
Similar right triangles showing sine and cosine of angle θ

In a right triangle with sidesa,b and hypotenusec,trigonometry determines thesine andcosine of the angleθ between sidea and the hypotenuse as:sin⁡θ=bc,cos⁡θ=ac.{\displaystyle \sin \theta ={\frac {b}{c}},\quad \cos \theta ={\frac {a}{c}}.}

From that it follows:cos2θ+sin2θ=a2+b2c2=1,{\displaystyle {\cos }^{2}\theta +{\sin }^{2}\theta ={\frac {a^{2}+b^{2}}{c^{2}}}=1,}where the last step applies Pythagoras's theorem. This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity.[38] In similar triangles, the ratios of the sides are the same regardless of the size of the triangles, and depend upon the angles. Consequently, in the figure, the triangle with hypotenuse of unit size has opposite side of sizesinθ and adjacent side of sizecosθ in units of the hypotenuse.

Relation to the cross product

The area of a parallelogram as a cross product; vectorsa andb identify a plane anda ×b is normal to this plane.

The Pythagorean theorem relates thecross product anddot product in a similar way:[39]‖a×b‖2+(a⋅b)2=‖a‖2‖b‖2.{\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}+(\mathbf {a} \cdot \mathbf {b} )^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2}.}

This can be seen from the definitions of the cross product and dot product, asa×b=abnsin⁡θa⋅b=abcos⁡θ,{\displaystyle {\begin{aligned}\mathbf {a} \times \mathbf {b} &=ab\,\mathbf {n} \sin {\theta }\\\mathbf {a} \cdot \mathbf {b} &=ab\cos {\theta },\end{aligned}}}withn aunit vector normal to botha andb. The relationship follows from these definitions and the Pythagorean trigonometric identity.

This can also be used to define the cross product. By rearranging the following equation is obtained‖a×b‖2=‖a‖2‖b‖2−(a⋅b)2.{\displaystyle \|\mathbf {a} \times \mathbf {b} \|^{2}=\|\mathbf {a} \|^{2}\|\mathbf {b} \|^{2}-(\mathbf {a} \cdot \mathbf {b} )^{2}.}

This can be considered as a condition on the cross product and so part of its definition, for example inseven dimensions.[40][41]

As an axiom

Main article:Parallel postulate

If the first four of theEuclidean geometry axioms are assumed to be true then the Pythagorean theorem is equivalent to the fifth. That is,Euclid's fifth postulate implies the Pythagorean theorem and vice-versa.

Generalizations

Similar figures on the three sides

The Pythagorean theorem generalizes beyond the areas of squares on the three sides to anysimilar figures. This was known byHippocrates of Chios in the 5th century BC,[42] and was included byEuclid in hisElements:[43]

If one erects similar figures (seeEuclidean geometry) with corresponding sides on the sides of a right triangle, then the sum of the areas of the ones on the two smaller sides equals the area of the one on the larger side.

This extension assumes that the sides of the original triangle are the corresponding sides of the three congruent figures (so the common ratios of sides between the similar figures area : b : c).[44] While Euclid's proof only applied to convex polygons, the theorem also applies to concave polygons and even to similar figures that have curved boundaries (but still with part of a figure's boundary being the side of the original triangle).[44]

The basic idea behind this generalization is that the area of a plane figure isproportional to the square of any linear dimension, and in particular is proportional to the square of the length of any side. Thus, if similar figures with areasA,B andC are erected on sides with corresponding lengthsa,b andc then:Aa2=Bb2=Cc2⟹A+B=a2c2C+b2c2C.{\displaystyle {\begin{aligned}&{\frac {A}{a^{2}}}={\frac {B}{b^{2}}}={\frac {C}{c^{2}}}\\&\quad \implies A+B={\frac {a^{2}}{c^{2}}}C+{\frac {b^{2}}{c^{2}}}C.\end{aligned}}}

But, by the Pythagorean theorem,a2 +b2 =c2, soA +B =C.

Conversely, if we can prove thatA +B =C for three similar figures without using the Pythagorean theorem, then we can work backwards to construct a proof of the theorem. For example, the starting center triangle can be replicated and used as a triangleC on its hypotenuse, and two similar right triangles (A andB ) constructed on the other two sides, formed by dividing the central triangle by itsaltitude. The sum of the areas of the two smaller triangles therefore is that of the third, thusA +B =C and reversing the above logic leads to the Pythagorean theorema2 +b2 =c2. (See alsoEinstein's proof by dissection without rearrangement)

Generalization for similar triangles,
green areaA +B = blue areaC
Pythagoras's theorem using similar right triangles
Generalization for regular pentagons

Law of cosines

Main article:Law of cosines
The separations of two points(r1,θ1) and(r2,θ2) inpolar coordinates is given by thelaw of cosines. Interior angleΔθ =θ1 −θ2.

The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines, which states thata2+b2−2abcos⁡θ=c2{\displaystyle a^{2}+b^{2}-2ab\cos {\theta }=c^{2}}whereθ{\displaystyle \theta } is the angle between sidesa{\displaystyle a} and⁠b{\displaystyle b}⁠.[45]

Whenθ{\displaystyle \theta } isπ2{\displaystyle {\tfrac {\pi }{2}}} radians or 90°, then⁠cos⁡θ=0{\displaystyle \cos {\theta }=0}⁠, and the formula reduces to the usual Pythagorean theorem.

Arbitrary triangle

Generalization of Pythagoras's theorem byTâbit ibn Qorra.[46] Lower panel: reflection of triangleCAD (top) to form triangleDAC, similar to triangleABC (top).

At any selected angle of a general triangle of sidesa,b,c, inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angleθ is opposite the side labeledc. Inscribing the isosceles triangle forms triangleCAD with angleθ opposite sideb and with sider alongc. A second triangle is formed with angleθ opposite sidea and a side with lengths alongc, as shown in the figure.Thābit ibn Qurra stated that the sides of the three triangles were related as:[47][48]a2+b2=c(r+s).{\displaystyle a^{2}+b^{2}=c(r+s).}As the angleθ approachesπ/2, the base of the isosceles triangle narrows, and lengthsr ands overlap less and less. Whenθ =π/2,ADB becomes a right triangle,r +s =c, and the original Pythagorean theorem is regained.

One proof observes that triangleABC has the same angles as triangleCAD, but in opposite order. (The two triangles share the angle at vertexA, both contain the angleθ, and so also have the same third angle by thetriangle postulate.) Consequently,ABC is similar to the reflection ofCAD, the triangleDAC in the lower panel. Taking the ratio of sides opposite and adjacent toθ,cb=br.{\displaystyle {\frac {c}{b}}={\frac {b}{r}}.}Likewise, for the reflection of the other triangle,ca=as.{\displaystyle {\frac {c}{a}}={\frac {a}{s}}.}Clearing fractions and adding these two relations:cs+cr=a2+b2,{\displaystyle cs+cr=a^{2}+b^{2},}the required result.

The theorem remains valid if the angleθ is obtuse so the lengthsr ands are non-overlapping.

General triangles using parallelograms

Generalization for arbitrary triangles,
green area = blue area
Construction for proof of parallelogram generalization

Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the area of the parallelogram on the longest side is the sum of the areas of the parallelograms on the other two sides, provided the parallelogram on the long side is constructed as indicated (the dimensions labeled with arrows are the same, and determine the sides of the bottom parallelogram). This replacement of squares with parallelograms bears a clear resemblance to the original Pythagoras's theorem, and was considered a generalization byPappus of Alexandria in 4 AD[49][50]

The lower figure shows the elements of the proof. Focus on the left side of the figure. The left green parallelogram has the same area as the left, blue portion of the bottom parallelogram because both have the same baseb and heighth. However, the left green parallelogram also has the same area as the left green parallelogram of the upper figure, because they have the same base (the upper left side of the triangle) and the same height normal to that side of the triangle. Repeating the argument for the right side of the figure, the bottom parallelogram has the same area as the sum of the two green parallelograms.

Solid geometry

Pythagoras's theorem in three dimensions relates the diagonal AD to the three sides.
A tetrahedron with outward facing right-angle corner

In terms ofsolid geometry, Pythagoras's theorem can be applied to three dimensions as follows. Consider thecuboid shown in the figure. The length offace diagonalAC is found from Pythagoras's theorem as:AC¯2=AB¯2+BC¯2,{\displaystyle {\overline {AC}}^{\,2}={\overline {AB}}^{\,2}+{\overline {BC}}^{\,2},}

where these three sides form a right triangle. Using diagonalAC and the horizontal edgeCD, the length ofbody diagonalAD then is found by a second application of Pythagoras's theorem as:AD¯2=AC¯2+CD¯2,{\displaystyle {\overline {AD}}^{\,2}={\overline {AC}}^{\,2}+{\overline {CD}}^{\,2},}or, doing it all in one step:AD¯2=AB¯2+BC¯2+CD¯2.{\displaystyle {\overline {AD}}^{\,2}={\overline {AB}}^{\,2}+{\overline {BC}}^{\,2}+{\overline {CD}}^{\,2}.}

This result is the three-dimensional expression for the magnitude of a vectorv (the diagonalAD) in terms of its orthogonal components{vk} (the three mutually perpendicular sides):‖v‖2=∑k=13‖vk‖2.{\displaystyle \|\mathbf {v} \|^{2}=\sum _{k=1}^{3}\|\mathbf {v} _{k}\|^{2}.}

This one-step formulation may be viewed as a generalization of Pythagoras's theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras's theorem to a succession of right triangles in a sequence of orthogonal planes.

A substantial generalization of the Pythagorean theorem to three dimensions isde Gua's theorem, named forJean Paul de Gua de Malves: If atetrahedron has a right angle corner (like a corner of acube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This result can be generalized as in the "n-dimensional Pythagorean theorem":[51]

Letx1,x2,…,xn{\displaystyle x_{1},x_{2},\ldots ,x_{n}} be orthogonal vectors inRn. Consider then-dimensional simplexS with vertices⁠0,x1,…,xn{\displaystyle 0,x_{1},\ldots ,x_{n}}⁠. (Think of the(n − 1)-dimensional simplex with verticesx1,…,xn{\displaystyle x_{1},\ldots ,x_{n}} not including the origin as the "hypotenuse" ofS and the remaining(n − 1)-dimensional faces ofS as its "legs".) Then the square of the volume of the hypotenuse ofS is the sum of the squares of the volumes of then legs.

This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed. The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras's theorem applies. In a different wording:[52]

Given ann-rectangularn-dimensional simplex, the square of the(n − 1)-content of thefacet opposing the right vertex will equal the sum of the squares of the(n − 1)-contents of the remaining facets.

Inner product spaces

Vectors involved in the parallelogram law

The Pythagorean theorem can be generalized toinner product spaces,[53] which are generalizations of the familiar 2-dimensional and 3-dimensionalEuclidean spaces. For example, afunction may be considered as avector with infinitely many components in an inner product space, as infunctional analysis.[54]

In an inner product space, the concept ofperpendicularity is replaced by the concept oforthogonality: two vectorsv andw are orthogonal if their inner product⟨v,w⟩{\displaystyle \langle \mathbf {v} ,\mathbf {w} \rangle } is zero. Theinner product is a generalization of thedot product of vectors. The dot product is called thestandard inner product or theEuclidean inner product. However, other inner products are possible.[55]

The concept of length is replaced by the concept of thenorm‖v‖ of a vectorv, defined as:[56]‖v‖≡⟨v,v⟩|.{\displaystyle \lVert \mathbf {v} \rVert \equiv {\textstyle {\sqrt {\langle \mathbf {v} ,\,\mathbf {v} \rangle }}{\vphantom {\big |}}}.}

In an inner-product space, thePythagorean theorem states that for any two orthogonal vectorsv andw we have‖v+w‖2=‖v‖2+‖w‖2.{\displaystyle \left\|\mathbf {v} +\mathbf {w} \right\|^{2}=\left\|\mathbf {v} \right\|^{2}+\left\|\mathbf {w} \right\|^{2}.}Here the vectorsv andw are akin to the sides of a right triangle with hypotenuse given by thevector sumv +w. This form of the Pythagorean theorem is a consequence of theproperties of the inner product:‖v+w‖2=⟨v+w, v+w⟩=⟨v,v⟩+⟨w,w⟩+⟨v,w⟩+⟨w,v⟩=‖v‖2+‖w‖2,{\displaystyle {\begin{aligned}\left\|\mathbf {v} +\mathbf {w} \right\|^{2}&=\langle \mathbf {v+w} ,\ \mathbf {v+w} \rangle \\[3mu]&=\langle \mathbf {v} ,\,\mathbf {v} \rangle +\langle \mathbf {w} ,\,\mathbf {w} \rangle +\langle \mathbf {v,\,w} \rangle +\langle \mathbf {w,\,v} \rangle \\[3mu]&=\left\|\mathbf {v} \right\|^{2}+\left\|\mathbf {w} \right\|^{2},\end{aligned}}}where⟨v,w⟩=⟨w,v⟩=0{\displaystyle \langle \mathbf {v,\,w} \rangle =\langle \mathbf {w,\,v} \rangle =0} because of orthogonality.

A further generalization of the Pythagorean theorem in an inner product space to non-orthogonal vectors is theparallelogram law:[56]2‖v‖2+2‖w‖2=‖v+w‖2+‖v−w‖2,{\displaystyle 2\|\mathbf {v} \|^{2}+2\|\mathbf {w} \|^{2}=\|\mathbf {v+w} \|^{2}+\|\mathbf {v-w} \|^{2},}

which says that twice the sum of the squares of the lengths of the sides of a parallelogram is the sum of the squares of the lengths of the diagonals. Any norm that satisfies this equality isipso facto a norm corresponding to an inner product.[56]

The Pythagorean identity can be extended to sums of more than two orthogonal vectors. Ifv1,v2, ...,vn are pairwise-orthogonal vectors in an inner-product space, then application of the Pythagorean theorem to successive pairs of these vectors (as described for 3-dimensions in the section onsolid geometry) results in the equation[57]‖∑k=1nvk‖2=∑k=1n‖vk‖2{\displaystyle {\biggl \|}\sum _{k=1}^{n}\mathbf {v} _{k}{\biggr \|}^{2}=\sum _{k=1}^{n}\|\mathbf {v} _{k}\|^{2}}

Sets ofm-dimensional objects inn-dimensional space

Another generalization of the Pythagorean theorem applies toLebesgue-measurable sets of objects in any number of dimensions. Specifically, the square of the measure of anm-dimensional set of objects in one or more parallelm-dimensionalflats inn-dimensionalEuclidean space is equal to the sum of the squares of the measures of theorthogonal projections of the object(s) onto allm-dimensional coordinate subspaces.[58]

In mathematical terms:μms2=∑i=1xμmpi2{\displaystyle \mu _{ms}^{2}=\sum _{i=1}^{x}\mu _{mp_{i}}^{2}}where:

  • μm{\displaystyle \mu _{m}} is a measure inm-dimensions (a length in one dimension, an area in two dimensions, a volume in three dimensions, etc.).
  • s{\displaystyle s} is a set of one or more non-overlappingm-dimensional objects in one or more parallelm-dimensional flats inn-dimensional Euclidean space.
  • μms{\displaystyle \mu _{ms}} is the total measure (sum) of the set ofm-dimensional objects.
  • p{\displaystyle p} represents anm-dimensional projection of the original set onto an orthogonal coordinate subspace.
  • μmpi{\displaystyle \mu _{mp_{i}}} is the measure of them-dimensional set projection ontom-dimensional coordinate subspacei. Because object projections can overlap on a coordinate subspace, the measure of each object projection in the set must be calculated individually, then measures of all projections added together to provide the total measure for the set of projections on the given coordinate subspace.
  • x{\displaystyle x} is the number of orthogonal,m-dimensional coordinate subspaces inn-dimensional space (Rn) onto which them-dimensional objects are projected(m ≤n):

x=(nm)=n!m!(n−m)!{\displaystyle x={\binom {n}{m}}={\frac {n!}{m!(n-m)!}}}

Non-Euclidean geometry

The Pythagorean theorem is derived from theaxioms ofEuclidean geometry, and in fact, were the Pythagorean theorem to fail for some right triangle, then the plane in which this triangle is contained cannot be Euclidean. More precisely, the Pythagorean theoremimplies, and is implied by, Euclid's Parallel (Fifth) Postulate.[59][60] Thus, right triangles in anon-Euclidean geometry[61] do not satisfy the Pythagorean theorem. For example, inspherical geometry, all three sides of the right triangle (saya,b, andc) bounding an octant of the unit sphere have length equal toπ/2, and all its angles are right angles, which violates the Pythagorean theorem becausea2 +b2 = 2c2 >c2.

Here two cases of non-Euclidean geometry are considered—spherical geometry andhyperbolic plane geometry; in each case, as in the Euclidean case for non-right triangles, the result replacing the Pythagorean theorem follows from the appropriate law of cosines.

However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition that the triangle be right is replaced with the condition that two of the angles sum to the third, sayA +B =C. The sides are then related as follows: the sum of the areas of the circles with diametersa andb equals the area of the circle with diameterc.[62]

Spherical geometry

Spherical triangle

For any righttriangle on a sphere of radiusR (for example, ifγ in the figure is a right angle), with sidesa,b, andc, the relation between the sides takes the form:[63]cos⁡cR=cos⁡aRcos⁡bR.{\displaystyle \cos {\frac {c}{R}}=\cos {\frac {a}{R}}\,\cos {\frac {b}{R}}.}

This equation can be derived as a special case of thespherical law of cosines that applies to all spherical triangles:cos⁡cR=cos⁡aRcos⁡bR+sin⁡aRsin⁡bRcos⁡γ.{\displaystyle \cos {\frac {c}{R}}=\cos {\frac {a}{R}}\,\cos {\frac {b}{R}}+\sin {\frac {a}{R}}\,\sin {\frac {b}{R}}\,\cos {\gamma }.}

For infinitesimal triangles on the sphere (or equivalently, for finite spherical triangles on a sphere of infinite radius), the spherical relation between the sides of a right triangle reduces to the Euclidean form of the Pythagorean theorem. To see how, assume we have a spherical triangle of fixed side lengthsa,b, andc on a sphere with expanding radiusR. AsR approaches infinity the quantitiesa/R,b/R, andc/R tend to zero and the spherical Pythagorean identity reduces to{{{1}}}, so we must look at itsasymptotic expansion.

TheMaclaurin series for the cosine function can be written ascos⁡x=1−12x2+O(x4){\textstyle \cos x=1-{\tfrac {1}{2}}x^{2}+O{\left(x^{4}\right)}} with the remainder term inbig O notation. Lettingx=c/R{\displaystyle x=c/R} be a side of the triangle, and treating the expression as an asymptotic expansion in terms ofR for a fixedc,cos⁡cR=1−c22R2+O(R−4){\displaystyle {\begin{aligned}\cos {\frac {c}{R}}=1-{\frac {c^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}\end{aligned}}}

and likewise fora andb. Substituting the asymptotic expansion for each of the cosines into the spherical relation for a right triangle yields1−c22R2+O(R−4)=(1−a22R2+O(R−4))(1−b22R2+O(R−4))=1−a22R2−b22R2+O(R−4).{\displaystyle {\begin{aligned}1-{\frac {c^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}&=\left(1-{\frac {a^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}\right)\left(1-{\frac {b^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}\right)\\&=1-{\frac {a^{2}}{2R^{2}}}-{\frac {b^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}.\end{aligned}}}

Subtracting 1 and then negating each side,c22R2=a22R2+b22R2+O(R−4).{\displaystyle {\frac {c^{2}}{2R^{2}}}={\frac {a^{2}}{2R^{2}}}+{\frac {b^{2}}{2R^{2}}}+O{\left(R^{-4}\right)}.}

Multiplying through by2R2, the asymptotic expansion forc in terms of fixeda,b and variableR isc2=a2+b2+O(R−2).{\displaystyle c^{2}=a^{2}+b^{2}+O{\left(R^{-2}\right)}.}

The Euclidean Pythagorean relationshipc2=a2+b2{\textstyle c^{2}=a^{2}+b^{2}} is recovered in the limit, as the remainder vanishes when the radiusR approaches infinity.

For practical computation in spherical trigonometry with small right triangles, cosines can be replaced with sines using the double-angle identitycos⁡2θ=1−2sin2⁡θ{\displaystyle \cos {2\theta }=1-2\sin ^{2}{\theta }} to avoidloss of significance. Then the spherical Pythagorean theorem can alternately be written assin2⁡c2R=sin2⁡a2R+sin2⁡b2R−2sin2⁡a2Rsin2⁡b2R.{\displaystyle \sin ^{2}{\frac {c}{2R}}=\sin ^{2}{\frac {a}{2R}}+\sin ^{2}{\frac {b}{2R}}-2\sin ^{2}{\frac {a}{2R}}\,\sin ^{2}{\frac {b}{2R}}.}

Hyperbolic geometry

Hyperbolic triangle

In ahyperbolic space with uniformGaussian curvature−1/R2, for a righttriangle with legsa,b, and hypotenusec, the relation between the sides takes the form:[64]cosh⁡cR=cosh⁡aRcosh⁡bR{\displaystyle \cosh {\frac {c}{R}}=\cosh {\frac {a}{R}}\,\cosh {\frac {b}{R}}}

where cosh is thehyperbolic cosine. This formula is a special form of thehyperbolic law of cosines that applies to all hyperbolic triangles:[65]cosh⁡cR=cosh⁡aRcosh⁡bR−sinh⁡aRsinh⁡bRcos⁡γ,{\displaystyle \cosh {\frac {c}{R}}=\cosh {\frac {a}{R}}\,\cosh {\frac {b}{R}}-\sinh {\frac {a}{R}}\,\sinh {\frac {b}{R}}\,\cos \gamma ,}withγ the angle at the vertex opposite the sidec.

By using theMaclaurin series for the hyperbolic cosine,coshx ≈ 1 +x2/2, it can be shown that as a hyperbolic triangle becomes very small (that is, asa,b, andc all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras's theorem.

For small right triangles(a,b ≪R), the hyperbolic cosines can be eliminated to avoidloss of significance, givingsinh2⁡c2R=sinh2⁡a2R+sinh2⁡b2R+2sinh2⁡a2Rsinh2⁡b2R.{\displaystyle \sinh ^{2}{\frac {c}{2R}}=\sinh ^{2}{\frac {a}{2R}}+\sinh ^{2}{\frac {b}{2R}}+2\sinh ^{2}{\frac {a}{2R}}\sinh ^{2}{\frac {b}{2R}}.}

Very small triangles

For any uniform curvatureK (positive, zero, or negative), in very small right triangles (|K|a2, |K|b2 ≪ 1) with hypotenusec, it can be shown thatc2=a2+b2−K3a2b2−K245a2b2(a2+b2)−2K3945a2b2(a2−b2)2+O(K4c10).{\displaystyle c^{2}=a^{2}+b^{2}-{\frac {K}{3}}a^{2}b^{2}-{\frac {K^{2}}{45}}a^{2}b^{2}(a^{2}+b^{2})-{\frac {2K^{3}}{945}}a^{2}b^{2}(a^{2}-b^{2})^{2}+O(K^{4}c^{10}).}

Differential geometry

Distance between infinitesimally separated points inCartesian coordinates (top) andpolar coordinates (bottom), as given by Pythagoras's theorem

The Pythagorean theorem applies toinfinitesimal triangles seen indifferential geometry. In three dimensional space, the distance between two infinitesimally separated points satisfiesds2=dx2+dy2+dz2,{\displaystyle ds^{2}=dx^{2}+dy^{2}+dz^{2},}

withds the element of distance and (dx,dy,dz) the components of the vector separating the two points. Such a space is called aEuclidean space. However, inRiemannian geometry, a generalization of this expression useful for general coordinates (not just Cartesian) and general spaces (not just Euclidean) takes the form:[66]ds2=∑i,jngijdxidxj{\displaystyle ds^{2}=\sum _{i,j}^{n}g_{ij}\,dx_{i}\,dx_{j}}

which is called themetric tensor.[b] It may be a function of position, and often describescurved space. A simple example is Euclidean (flat) space expressed incurvilinear coordinates. For example, inpolar coordinates:ds2=dr2+r2dθ2.{\displaystyle ds^{2}=dr^{2}+r^{2}d\theta ^{2}.}

History

ThePlimpton 322 tablet recordsPythagorean triples fromBabylonian times.[67]

There is debate whether the Pythagorean theorem was discovered once, or many times in many places, and the date of first discovery is uncertain, as is the date of the first proof. The history of the theorem can be divided into four parts: knowledge ofPythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within somedeductive system.

Writtenc. 1800 BC, theEgyptianMiddle KingdomBerlin Papyrus 6619 includes a problem whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. According toPlutarch, the ancient Egyptians did know about the 3:4:5 right triangle, identifying its sides withOsiris,Isis, andHorus respectively.[68]

Historians ofMesopotamian mathematics have concluded that the Pythagorean rule was in widespread use during theOld Babylonian period (20th to 16th centuries BC), over a thousand years beforePythagoras was born.[69][70][71][72] The Mesopotamian tabletPlimpton 322, written nearLarsa alsoc. 1800 BC, contains entries that can be interpreted as the sides and diagonals of 15 different Pythagorean triples.[73] Another tablet from a similar time,YBC 7289, calculates the diagonal of a square or, equivalently, or an isosceles right triangle.[74]

InIndia, theBaudhayanaShulba Sutra, the dates of which are given variously as between the 8th and 5th century BC,[75] contains a list of Pythagorean triples and a statement of the Pythagorean theorem, both in the special case of theisoscelesright triangle and in the general case, as does theApastamba Shulba Sutra (c. 600 BC).[c]

ByzantineNeoplatonic philosopher and mathematicianProclus, writing in the fifth century AD, states two arithmetic rules, "one of them attributed toPlato, the other to Pythagoras",[77] for generating special Pythagorean triples. The rule attributed to Pythagoras (c. 570 – c. 495 BC) starts from anodd number and produces a triple with leg and hypotenuse differing by one unit; the rule attributed to Plato (428/427 or 424/423 – 348/347 BC) starts from an even number and produces a triple with leg and hypotenuse differing by two units. According toThomas L. Heath (1861–1940), no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived.[78] However, when authors such asPlutarch andCicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted.[79][80]ClassicistKurt von Fritz wrote, "Whether this formula is rightly attributed to Pythagoras personally ... one can safely assume that it belongs to the very oldest period ofPythagorean mathematics."[81] Around 300 BC, in Euclid'sElements, the oldest extantaxiomatic proof of the theorem is presented.[82]

Geometric proof of the Pythagorean theorem from theZhoubi Suanjing

With contents known much earlier, but in surviving texts dating from roughly the 1st century BC, theChinese textZhoubi Suanjing (周髀算经), (The Arithmetical Classic of theGnomon and the Circular Paths of Heaven) gives a reasoning for the Pythagorean theorem for the (3, 4, 5) triangle — in China it is called the "Gougu theorem" (勾股定理).[83][84] During theHan Dynasty (202 BC to 220 AD), Pythagorean triples appear inThe Nine Chapters on the Mathematical Art,[85] together with a mention of right triangles.[86] Some believe the theorem arose first inChina in the 11th century BC,[87] where it is alternatively known as the "Shang Gao theorem" (商高定理),[88] named after theDuke of Zhou's astronomer and mathematician, whose reasoning composed most of what was in theZhoubi Suanjing.[89]

See also

Notes and references

Notes

  1. ^The proof by Pythagoras probably was not a general one, as the theory of proportions was developed only two centuries after Pythagoras; seeMaor (2007), p. 25.
  2. ^Sometimes, by abuse of language, the same term is applied to the set of coefficientsgij.
  3. ^van der Waerden (1983), p. 26 believed that this material "was certainly based on earlier traditions".Carl Boyer states that the Pythagorean theorem in theŚulba-sũtram may have been influenced by ancient Mesopotamian math, but there is no conclusive evidence in favor or opposition of this possibility.[76]

References

  1. ^Saikia 2013.
  2. ^abSally & Sally (2007), p. 63, Chapter 3: Pythagorean triples.
  3. ^Benson, Donald (1999).The Moment of Proof: Mathematical Epiphanies. Oxford University Press. pp. 172–173.
  4. ^Euclid 1956, pp. 351–352.
  5. ^Huffman, Carl (23 February 2005)."Pythagoras". InZalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy (Winter 2018 Edition) (Winter 2018 ed.).It should now be clear that decisions about sources are crucial in addressing the question of whether Pythagoras was a mathematician and scientist. The view of Pythagoras's cosmos sketched in the first five paragraphs of this section, according to which he was neither a mathematician nor a scientist, remains the consensus.
  6. ^Bogomolny (2016),Proof #4.
  7. ^Bogomolny (2016),Proof #3.
  8. ^Loomis 1940.
  9. ^Maor (2007), p. 39.
  10. ^Schroeder, Manfred Robert (2012).Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. Courier Corporation. pp. 3–4.ISBN 978-0486134789.
  11. ^See for examplePythagorean theorem by shear mappingArchived 2016-10-14 at theWayback Machine, Saint Louis University website Java applet
  12. ^Gullberg, Jan (1997).Mathematics: from the birth of numbers. W. W. Norton & Company. p. 435.ISBN 0-393-04002-X.
  13. ^Heiberg, J. L."Euclid's Elements of Geometry"(PDF). pp. 46–47.
  14. ^"Euclid's Elements, Book I, Proposition 47". See also aweb page version using Java applets byDavid E. Joyce, Clark University.
  15. ^Hawking (2005), p. 12. This proof first appeared after a computer program was set to check Euclidean proofs.
  16. ^Bogomolny (2016),Proof #10.
  17. ^(Loomis 1940, p. 113, Geometric proof 22 and Figure 123)
  18. ^Polster, Burkard (2004).Q.E.D.: Beauty in Mathematical Proof. Walker Publishing Company. p. 49.
  19. ^Published in a weekly mathematics column:Garfield, James A. (1876)."Pons Asinorum".The New England Journal of Education.3 (14): 161; as noted inDunham, William (1997).The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities. Wiley. p. 96.ISBN 0-471-17661-3.
  20. ^Lantz, David (2008)."Garfield's proof of the Pythagorean Theorem".Colgate University Faculty Pages.Archived from the original on 28 August 2013. Retrieved14 January 2018.
  21. ^Maor (2007), pp. 106–107.
  22. ^Staring, Mike (1996). "The Pythagorean proposition: A proof by means of calculus".Mathematics Magazine.69 (1). Mathematical Association of America:45–46.doi:10.2307/2691395.JSTOR 2691395.
  23. ^Bogomolny (2016),Proof #40 (by M. Hardy).
  24. ^Berndt, Bruce C. (1988). "Ramanujan – 100 years old (fashioned) or 100 years new (fangled)?".The Mathematical Intelligencer.10 (3):24–31.doi:10.1007/BF03026638.S2CID 123311054.
  25. ^Sally & Sally (2007), pp. 54–55, Theorem 2.4 (Converse of the Pythagorean theorem).
  26. ^Euclid's Elements, Book I, Proposition 48 FromD.E. Joyce's web page at Clark University
  27. ^Casey, Stephen, "The converse of the theorem of Pythagoras",Mathematical Gazette 92, July 2008, 309–313.
  28. ^Mitchell, Douglas W., "Feedback on 92.47",Mathematical Gazette 93, March 2009, 156.
  29. ^Wilczynski, Ernest Julius;Slaught, Herbert Ellsworth (1914). "Theorem 1 and Theorem 2".Plane Trigonometry and Applications. Allyn and Bacon. p. 85.
  30. ^Dijkstra, Edsger W. (7 September 1986)."On the theorem of Pythagoras".EWD975. E. W. Dijkstra Archive.
  31. ^Bogomolny, Alexander (2018)."Pythagorean Theorem for the Reciprocals".Cut the Knot. Archived fromthe original on 3 December 2024. Retrieved10 November 2025.
  32. ^Law, Henry (1853)."Corollary 5 of Proposition XLVII (Pythagoras's Theorem)".The Elements of Euclid: with many additional propositions, and explanatory notes, to which is prefixed an introductory essay on logic. John Weale. p. 49.
  33. ^Lavine, Shaughan (1994).Understanding the infinite. Harvard University Press. p. 13.ISBN 0-674-92096-1.
  34. ^Heath (1921), Vol I, pp. 65; Hippasus was on a voyage at the time, and his fellows cast him overboard. SeeChoike, James R. (1980). "The Pentagram and the Discovery of an Irrational Number".The College Mathematics Journal.11:312–316.
  35. ^Fritz (1945).
  36. ^Jon Orwant; Jarkko Hietaniemi; John Macdonald (1999)."Euclidean distance".Mastering algorithms with Perl. O'Reilly Media, Inc. p. 426.ISBN 1-56592-398-7.
  37. ^Wentworth & Smith (1914), p. 116: "the Law of Cosines may be stated as follows: [...] In other words we have the Pythagorean Theorem as a special case. Hence this is sometimes called theGeneralized Pythagorean Theorem."
  38. ^Lawrence S. Leff (2005).PreCalculus the Easy Way (7th ed.). Barron's Educational Series. p. 296.ISBN 0-7641-2892-2.
  39. ^WS Massey (December 1983)."Cross products of vectors in higher-dimensional Euclidean spaces"(PDF).The American Mathematical Monthly.90 (10). Mathematical Association of America:697–701.doi:10.2307/2323537.JSTOR 2323537.S2CID 43318100. Archived fromthe original(PDF) on 26 February 2021.
  40. ^Pertti Lounesto (2001)."§7.4 Cross product of two vectors".Clifford algebras and spinors (2nd ed.). Cambridge University Press. p. 96.ISBN 0-521-00551-5.
  41. ^Francis Begnaud Hildebrand (1992).Methods of applied mathematics (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover Publications. p. 24.ISBN 0-486-67002-3.
  42. ^Heath, T. L.,A History of Greek Mathematics, Oxford University Press, 1921; reprinted by Dover, 1981.
  43. ^Euclid'sElements: Book VI, Proposition VI 31: "In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle."
  44. ^abPutz, John F.; Sipka, Timothy A. (September 2003). "On generalizing the Pythagorean theorem".The College Mathematics Journal.34 (4):291–295.
  45. ^Lawrence S. Leff (1 May 2005).cited work. Barron's Educational Series. p. 326.ISBN 0-7641-2892-2.
  46. ^Howard Whitley Eves (1983). "§4.8:...generalization of Pythagorean theorem".Great moments in mathematics (before 1650). Mathematical Association of America. p. 41.ISBN 0-88385-310-8.
  47. ^Sayili, Aydin (March 1960). "Thâbit ibn Qurra's Generalization of the Pythagorean Theorem".Isis.51 (1):35–37.doi:10.1086/348837.JSTOR 227603.S2CID 119868978.
  48. ^Sally & Sally (2007), p. 62, Exercise 2.10 (ii).
  49. ^For the details of such a construction, seeJennings, George (1997). "Figure 1.32: The generalized Pythagorean theorem".Modern geometry with applications: with 150 figures (3rd ed.). Springer. p. 23.ISBN 0-387-94222-X.
  50. ^Claudi Alsina, Roger B. Nelsen:Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010,ISBN 9780883853481, pp. 77–78 (excerpt, p. 77, atGoogle Books)
  51. ^Rajendra Bhatia (1997).Matrix analysis. Springer. p. 21.ISBN 0-387-94846-5.
  52. ^For an extended discussion of this generalization, see, for example,Willie W. WongArchived 2009-12-29 at theWayback Machine 2002,A generalized n-dimensional Pythagorean theorem.
  53. ^Ferdinand van der Heijden; Dick de Ridder (2004).Classification, parameter estimation, and state estimation. Wiley. p. 357.ISBN 0-470-09013-8.
  54. ^Qun Lin; Jiafu Lin (2006).Finite element methods: accuracy and improvement. Elsevier. p. 23.ISBN 7-03-016656-6.
  55. ^Howard Anton; Chris Rorres (2010).Elementary Linear Algebra: Applications Version (10th ed.). Wiley. p. 336.ISBN 978-0-470-43205-1.
  56. ^abcKaren Saxe (2002)."Theorem 1.2".Beginning functional analysis. Springer. p. 7.ISBN 0-387-95224-1.
  57. ^Douglas, Ronald G. (1998).Banach Algebra Techniques in Operator Theory (2nd ed.). New York, New York: Springer-Verlag New York, Inc. pp. 60–61.ISBN 978-0-387-98377-6.
  58. ^Donald R Conant & William A Beyer (March 1974). "Generalized Pythagorean Theorem".The American Mathematical Monthly.81 (3). Mathematical Association of America:262–265.doi:10.2307/2319528.JSTOR 2319528.
  59. ^Weisstein, Eric W. (1999)."Parallel Postulate".CRC Concise Encyclopedia of Mathematics. CRC Press. p. 1313.ISBN 0-8493-9640-9.The parallel postulate is equivalent to theEquidistance postulate,Playfair axiom,Proclus axiom, theTriangle postulate and thePythagorean theorem.
  60. ^Pruss, Alexander R. (2006).The Principle of Sufficient Reason: A Reassessment. Cambridge University Press. p. 11.ISBN 0-521-85959-X.We could include ... the parallel postulate and derive the Pythagorean theorem. Or we could instead make the Pythagorean theorem among the other axioms and derive the parallel postulate.
  61. ^Hawking (2005), p. 4.
  62. ^Pambuccian, Victor (December 2010)."Maria Teresa Calapso's Hyperbolic Pythagorean Theorem".The Mathematical Intelligencer.32 (4): 2.doi:10.1007/s00283-010-9169-0.
  63. ^O'Neill, Barrett (2006)."Exercise 4".Elementary Differential Geometry (2nd ed.). Academic Press. p. 441.ISBN 0-12-088735-5.
  64. ^Stahl, Saul (1993)."Theorem 8.3".The Poincaré Half-Plane: A Gateway to Modern Geometry. Jones & Bartlett Learning. p. 122.ISBN 0-86720-298-X.
  65. ^Gilman, Jane (1995)."Hyperbolic triangles".Two-generator Discrete Subgroups of PSL(2,R). American Mathematical Society.ISBN 0-8218-0361-1.
  66. ^Chow, Tai L. (2000).Mathematical methods for physicists: a concise introduction. Cambridge University Press. p. 52.ISBN 0-521-65544-7.
  67. ^Neugebauer 1969, p. 36.
  68. ^Plutarch (1936).Moralia V: Isis and Osiris. Loeb Classical Library. Vol. 306. Translated by Babbitt, Frank Cole. Harvard University Press. p. 135.
  69. ^Neugebauer 1969: p. 36 "In other words it was known during the whole duration of Babylonian mathematics that the sum of the squares on the lengths of the sides of a right triangle equals the square of the length of the hypotenuse."
  70. ^Friberg, Jöran (1981)."Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations".Historia Mathematica.8:277–318.doi:10.1016/0315-0860(81)90069-0.: p. 306 "Although Plimpton 322 is a unique text of its kind, there are several other known texts testifying that the Pythagorean theorem was well known to the mathematicians of the Old Babylonian period."
  71. ^Høyrup, Jens (1999). "Pythagorean 'Rule' and 'Theorem' – Mirror of the Relation Between Babylonian and Greek Mathematics". In Renger, Johannes (ed.).Babylon: Focus mesopotamischer Geschichte, Wiege früher Gelehrsamkeit, Mythos in der Moderne. 2. Internationales Colloquium der Deutschen Orient-Gesellschaft 24.–26. März 1998 in Berlin(PDF). Berlin: Deutsche Orient-Gesellschaft / Saarbrücken: SDV Saarbrücker Druckerei und Verlag. pp. 393–407., p. 406, "To judge from this evidence alone it is therefore likely that the Pythagorean rule was discovered within the lay surveyors' environment, possibly as a spin-off from the problem treated in Db2-146, somewhere between 2300 and 1825 BC." (Db2-146 is an Old Babylonian clay tablet fromEshnunna concerning the computation of the sides of a rectangle given its area and diagonal.)
  72. ^Robson 2008, p. 109: "Many Old Babylonian mathematical practitioners ... knew that the square on the diagonal of a right triangle had the same area as the sum of the squares on the length and width: that relationship is used in the worked solutions to word problems on cut-and-paste 'algebra' on seven different tablets, from Ešnuna, Sippar, Susa, and an unknown location in southern Babylonia."
  73. ^Robson 2001.
  74. ^Mackinnon, Nick (March 1992). "Homage to Babylonia".The Mathematical Gazette.76 (475):158–178.doi:10.2307/3620389.JSTOR 3620389.
  75. ^Kim Plofker (2009).Mathematics in India. Princeton University Press. pp. 17–18.ISBN 978-0-691-12067-6.
  76. ^Boyer & Merzbach (2011), p. 187: "[In Sulba-sutras,] we find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. Although Mesopotamian influence in theSulvasũtras is not unlikely, we know of no conclusive evidence for or against this. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides. Less easily explained is another rule given by Apastamba – one that strongly resembles some of the geometric algebra in Book II of Euclid'sElements. (...)"
  77. ^Proclus (1970).A Commentary of the First Book of Euclid'sElements. Translated by Morrow, Glenn R. Princeton University Press. 428.6.
  78. ^"Introduction and books 1,2". The University Press. 25 March 1908 – via Google Books.
  79. ^(Heath 1921, Vol I, p. 144): "Though this is the proposition universally associated by tradition with the name of Pythagoras, no really trustworthy evidence exists that it was actually discovered by him. The comparatively late writers who attribute it to him add the story that he sacrificed an ox to celebrate his discovery."
  80. ^An extensive discussion of the historical evidence is provided in (Euclid 1956, p. 351)page=351
  81. ^Fritz (1945), p. 252.
  82. ^Aaboe, Asger (1997).Episodes From the Early History of Mathematics. Mathematical Association of America. p. 51.ISBN 0-88385-613-1.... it is not until Euclid that we find a logical sequence of general theorems with proper proofs.
  83. ^Crease, Robert P. (2008).The Great Equations: Breakthroughs in Science From Pythagoras to Heisenberg. W W Norton & Co. p. 25.ISBN 978-0-393-06204-5.
  84. ^A rather extensive discussion of the origins of the various texts in the Zhou Bi is provided byCullen, Christopher (2007).Astronomy and Mathematics in Ancient China: The 'Zhou Bi Suan Jing'. Cambridge University Press. pp. 139ff.ISBN 978-0-521-03537-8.
  85. ^This work is a compilation of 246 problems, some of which survived the book burning of 213 BC, and was put in final form before 100 AD. It was extensively commented upon by Liu Hui in 263 AD.Straffin, Philip D. Jr. (2004)."Liu Hui and the First Golden Age of Chinese Mathematics". In Anderson, Marlow; Katz, Victor J.; Wilson, Robin J. (eds.).Sherlock Holmes in Babylon: and Other Tales of Mathematical History. Mathematical Association of America. pp. 69ff.ISBN 0-88385-546-1. See particularly §3:Nine chapters on the mathematical art, pp. 71ff.
  86. ^Shen, Kangshen; Crossley, John N.; Lun, Anthony Wah-Cheung (1999).The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press. p. 488.ISBN 0-19-853936-3.
  87. ^In particular, Li Jimin; seeCentaurus, Volume 39. Copenhagen: Munksgaard. 1997. pp. 193, 205.
  88. ^Chen, Cheng-Yih (1996)."§3.3.4 Chén Zǐ's formula and the Chóng-Chã method; Figure 40".Early Chinese Work in Natural Science: a Re-examination of the Physics of Motion, Acoustics, Astronomy and Scientific Thoughts. Hong Kong University Press. p. 142.ISBN 962-209-385-X.
  89. ^Wu, Wen-tsün (2008)."The Gougu Theorem".Selected works of Wen-tsün Wu. World Scientific. p. 158.ISBN 978-981-279-107-8.

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