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Equilateral triangle

From Wikipedia, the free encyclopedia

Shape with three equal sides

"Equilateral" redirects here. For other uses, seeEquilateral (disambiguation).

Equilateral triangle

Type	Regular polygon
Edges andvertices	3
Schläfli symbol	{3}
Coxeter–Dynkin diagrams
Symmetry group	$\mathrm {D} _{3}$
Area	${\textstyle {\frac {\sqrt {3}}{4}}a^{2}}$
Internal angle (degrees)	60°

Anequilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is aregular polygon, occasionally known as theregular triangle. It is the special case of anisosceles triangle by modern definition, creating more special properties.

The equilateral triangle can be found in varioustilings, and inpolyhedrons such as thedeltahedron andantiprism. It appears in real life in popular culture, architecture, and the study ofstereochemistry resembling the molecular known as thetrigonal planar molecular geometry.

Properties

An equilateral triangle is a triangle that has three equal sides. It is a special case of anisosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides.^[1] Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base.^[2]

The follow-up definition above may result in more precise properties. For example, since theperimeter of an isosceles triangle is the sum of its two legs and base, the equilateral triangle is formulated as three times its side.^[3]^[4] Theinternal angles of an equilateral triangle are equal, 60°. Because of these properties, the equilateral triangles areregular polygons. Thecevians of an equilateral triangle are all equal in length, resulting in themedian andangle bisector being equal in length, considering those lines as their altitude depending on the base's choice.^[5] When the equilateral triangle is flipped across its altitude or rotated around its center for one-third of a full turn, its appearance is unchanged; it has the symmetry of adihedral group $\mathrm {D} _{3}$ oforder six.^[6] Other properties are discussed below.

Area

The right triangle with ahypotenuse of $1 {\displaystyle 1}$ has a height of ${\sqrt {3}}/2$ , the sine of 60°.

{\displaystyle 1} — The right triangle with ahypotenuse of $1 {\displaystyle 1}$ has a height of ${\sqrt {3}}/2$ , the sine of 60°.

The area of an equilateral triangle with edge length $a {\displaystyle a}$ is $T={\frac {\sqrt {3}}{4}}a^{2}.$ The formula may be derived from the formula of an isosceles triangle byPythagoras theorem: the altitude $h {\displaystyle h}$ of a triangle isthe square root of the difference of squares of a side and half of a base.^[3] Since the base and the legs are equal, the height is:^[7] $h={\sqrt {a^{2}-{\frac {a^{2}}{4}}}}={\frac {\sqrt {3}}{2}}a.$ In general, thearea of a triangle is half the product of its base and height. The formula for the area of an equilateral triangle can be obtained by substituting the altitude formula.^[7] Another way to prove the area of an equilateral triangle is by using thetrigonometric function. The area of a triangle is formulated as the half product of base and height and the sine of an angle. Because all of the angles of an equilateral triangle are 60°, the formula is as desired.^{[citation needed]}

A version of theisoperimetric inequality for triangles states that the triangle of greatestarea among all those with a givenperimeter is equilateral. That is, for perimeter $p {\displaystyle p}$ and area $T {\displaystyle T}$ , the equality holds for the equilateral triangle:^[8] $p^{2}=12{\sqrt {3}}T.$

Relationship with circles

The radius of thecircumscribed circle is: $R={\frac {a}{\sqrt {3}}},$ and the radius of theinscribed circle is half of the circumradius: $r={\frac {\sqrt {3}}{6}}a.$

Atheorem of Euler states that the distance $t {\displaystyle t}$ between circumcenter and incenter is formulated as $t^{2}=R(R-2r)$ . As a corollary of this, the equilateral triangle has the smallest ratio of the circumradius $R {\displaystyle R}$ to the inradius $r {\displaystyle r}$ of any triangle. That is:^[9] $R\geq 2r.$

Pompeiu's theorem states that, if $P {\displaystyle P}$ is an arbitrary point in the plane of an equilateral triangle $A B C {\displaystyle ABC}$ but not on itscircumcircle, then there exists a triangle with sides of lengths $P A {\displaystyle PA}$ , $P B {\displaystyle PB}$ , and $P C {\displaystyle PC}$ . That is, $P A {\displaystyle PA}$ , $P B {\displaystyle PB}$ , and $P C {\displaystyle PC}$ satisfy thetriangle inequality that the sum of any two of them is greater than the third. If $P {\displaystyle P}$ is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known asVan Schooten's theorem.^[10]

Apacking problem asks the objective of $n {\displaystyle n}$ circles packing into the smallest possible equilateral triangle. The optimal solutions show $n<13$ that can be packed into the equilateral triangle, but the open conjectures expand to $n<28$ .^[11]

Other mathematical properties

Visual proof of Viviani's theorem

Morley's trisector theorem states that, in any triangle, the three points of intersection of the adjacentangle trisectors form an equilateral triangle.

Viviani's theorem states that, for any interior point $P {\displaystyle P}$ in an equilateral triangle with distances $d {\displaystyle d}$ , $e {\displaystyle e}$ , and $f {\displaystyle f}$ from the sides and altitude $h {\displaystyle h}$ , $d+e+f=h,$ independent of the location of $P {\displaystyle P}$ .^[12]

An equilateral triangle may haveinteger sides with three rational angles as measured in degrees,^[13] known for the only acute triangle that is similar to itsorthic triangle (with vertices at the feet of thealtitudes),^[14] and the only triangle whoseSteiner inellipse is a circle (specifically, the incircle). The triangle of the largest area of all those inscribed in a given circle is equilateral, and the triangle of the smallest area of all those circumscribed around a given circle is also equilateral.^[15] It is the only regular polygon aside from thesquare that can beinscribed inside any other regular polygon.

Given a point $P {\displaystyle P}$ in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides is greater than or equal to 2, equality holding when $P {\displaystyle P}$ is the centroid. In no other triangle is there a point for which this ratio is as small as 2.^[16] This is theErdős–Mordell inequality; a stronger variant of it isBarrow's inequality, which replaces the perpendicular distances to the sides with the distances from $P {\displaystyle P}$ to the points where theangle bisectors of $\angle APB$ , $\angle BPC$ , and $\angle CPA$ cross the sides ( $A {\displaystyle A}$ , $B {\displaystyle B}$ , and $C {\displaystyle C}$ being the vertices). There are numerous othertriangle inequalities that hold equality if and only if the triangle is equilateral.

Construction

Construction of equilateral triangle with compass and straightedge

A regular polygon is constructible by compass and straightedge if and only if the odd prime factors of its number of sides are distinctFermat primes.^[17] There are five known Fermat primes: 3, 5, 17, 257, 65537.

The very first proposition in theElements byEuclid starts by drawing a circle with a certain radius, placing the point of the compass on the circle, and drawing another circle with the same radius; the two circles intersect in two points. An equilateral triangle can be constructed by joining the two centers of the circles and one of the points of intersection.^[18]

Equivalently, begin with anyline segment as one side; place the point of the compass on one end of the line, then swing an arc from that point to the other point of the line segment; repeat with the other side of the line, which connects the point where the two arcs intersect with each end of the line segment in the aftermath.

If three equilateral triangles are constructed on the sides of an arbitrary triangle, either all outward or inward, byNapoleon's theorem the centers of those equilateral triangles themselves form an equilateral triangle.

Appearances

In other related figures

The equilateral triangle tiling fills the plane

The Sierpiński triangle

Notably, the equilateral triangletiles theEuclidean plane with six triangles meeting at a vertex; the dual of this tessellation is thehexagonal tiling.Truncated hexagonal tiling,rhombitrihexagonal tiling,trihexagonal tiling,snub square tiling, andsnub hexagonal tiling are allsemi-regular tessellations constructed with equilateral triangles.^[19] Other two-dimensional objects built from equilateral triangles include theSierpiński triangle (afractal shape constructed from an equilateral triangle by subdividing recursively into smaller equilateral triangles) andReuleaux triangle (acurved triangle withconstant width, constructed from an equilateral triangle by rounding each of its sides).^[10]

The regular octahedron is adeltahedron, as well as a member of the family ofantiprisms.

Equilateral triangles may also form a polyhedron in three dimensions. A polyhedron whose faces are all equilateral triangles is called adeltahedron. There are eightstrictly convex deltahedra: three of the fivePlatonic solids (regular tetrahedron,regular octahedron, andregular icosahedron) and five of the 92Johnson solids (triangular bipyramid,pentagonal bipyramid,snub disphenoid,triaugmented triangular prism, andgyroelongated square bipyramid).^[20] More generally, allJohnson solids have equilateral triangles among their faces, though most also have otherregular polygons.^[21]

Theantiprisms are a family of polyhedra incorporating a band of alternating triangles. When the antiprism isuniform, its bases are regular and all triangular faces are equilateral.^[22]

As a generalization, the equilateral triangle belongs to the infinite family of $n {\displaystyle n}$ -simplexes, with $n=2$ .^[23]

Applications

Equilateral triangle usage as a yield sign

Equilateral triangles have frequently appeared in man-made constructions and in popular culture. In architecture, an example can be seen in the cross-section of theGateway Arch and the surface of theVegreville egg.^[24]^[25] It appears in theflag of Nicaragua and theflag of the Philippines.^[26]^[27] It is a shape of a variety ofroad signs, including theyield sign.^[28]

The equilateral triangle occurs in the study ofstereochemistry. It can be described as themolecular geometry in which one atom in the center connects three other atoms in a plane, known as thetrigonal planar molecular geometry.^[29]

In theThomson problem, concerning the minimum-energy configuration of $n {\displaystyle n}$ charged particles on a sphere, and for theTammes problem of constructing aspherical code maximizing the smallest distance among the points, the best solution known for $n=3$ places the points at the vertices of an equilateral triangle,inscribed in the sphere. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is unknown.^[30]

See also

References

Notes

^Stahl (2003), p. 37.
^Lardner (1840), p. 46.
^^a ^bHarris & Stocker (1998), p. 78.
^Cerin (2004), See Theorem 1.
^Owen, Felix & Deirdre (2010), p. 36, 39.
^Carstensen, Fine & Rosenberger (2011), p. 156.
^^a ^bMcMullin & Parkinson (1936), p. 96.
^Chakerian (1979).
^Svrtan & Veljan (2012).
^^a ^bAlsina & Nelsen (2010), p. 102–103.
^Melissen & Schuur (1995).
^Posamentier & Salkind (1996).
^Conway & Guy (1996), p. 201, 228–229.
^Bankoff & Garfunkel (1973), p. 19.
^Dörrie (1965), p. 379–380.
^Lee (2001).
^Křížek, Luca & Somer (2001), p. 1–2.
^Cromwell (1997), p. 62.
^Grünbaum & Shepard (1977).
^Trigg (1978).
^Berman (1971).
^Horiyama et al. (2015), p. 124.
^Coxeter (1948), p. 120–121.
^Pelkonen & Albrecht (2006), p. 160.
^Alsina & Nelsen (2015), p. 22.
^White & Calderón (2008), p. 3.
^Guillermo (2012), p. 161.
^Riley, Cochran & Ballard (1982).
^Petrucci, Harwood & Herring (2002), p. 413–414, See Table 11.1.
^Whyte (1952).

Works cited

Alsina, Claudi; Nelsen, Roger B. (2010).Charming Proofs: A Journey Into Elegant Mathematics.Mathematical Association of America.ISBN 9780883853481.
Alsina, Claudi; Nelsen, Roger B. (2015).A Mathematical Space Odyssey: Solid Geometry in the 21st Century. Vol. 50.Mathematical Association of America.ISBN 978-1-61444-216-5.
Bankoff, Leon; Garfunkel, Jack (January 1973). "The heptagonal triangle".Mathematics Magazine.46 (1):7–19.doi:10.1080/0025570X.1973.11976267.
Berman, Martin (1971). "Regular-faced convex polyhedra".Journal of the Franklin Institute.291 (5):329–352.doi:10.1016/0016-0032(71)90071-8.MR 0290245.
Cerin, Zvonko (2004)."The vertex-midpoint-centroid triangles"(PDF).Forum Geometricorum.4:97–109.
Carstensen, Celine; Fine, Celine; Rosenberger, Gerhard (2011).Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography.De Gruyter. p. 156.ISBN 978-3-11-025009-1.
Chakerian, G. D. (1979)."Chapter 7: A Distorted View of Geometry". In Honsberger, R. (ed.).Mathematical Plums. Washington DC:Mathematical Association of America. p. 147.
Conway, J. H.;Guy, R. K. (1996).The Book of Numbers. Springer-Verlag.
Coxeter, H. S. M. Coxeter (1948).Regular Polytopes (1 ed.). London:Methuen & Co. LTD.OCLC 4766401.Zbl 0031.06502.
Cromwell, Peter R. (1997).Polyhedra.Cambridge University Press.ISBN 978-0-521-55432-9.
Dörrie, Heinrich (1965).100 Great Problems of Elementary Mathematics.Dover Publications.
Grünbaum, Branko;Shepard, Geoffrey (November 1977)."Tilings by Regular Polygons"(PDF).Mathematics Magazine.50 (5). Taylor & Francis, Ltd.:231–234.doi:10.2307/2689529.JSTOR 2689529.MR 1567647.Zbl 0385.51006. Archived fromthe original(PDF) on 2016-03-03. Retrieved2023-03-09.
Guillermo, Artemio R. (2012).Historical Dictionary of the Philippines. Scarecrow Press.ISBN 978-0810872462.
Harris, John W.; Stocker, Horst (1998).Handbook of mathematics and computational science. New York: Springer-Verlag.ISBN 0-387-94746-9.MR 1621531.
Herz-Fischler, Roger (2000).The Shape of the Great Pyramid. Wilfrid Laurier University Press.ISBN 0-88920-324-5.
Horiyama, Takayama; Itoh, Jin-ichi; Katoh, Naoi; Kobayashi, Yuki; Nara, Chie (14–16 September 2015). "Continuous Folding of Regular Dodecahedra". In Akiyama, Jin; Ito, Hiro; Sakai, Toshinori; Uno, Yushi (eds.).Discrete and Computational Geometry and Graphs. Japanese Conference on Discrete and Computational Geometry and Graphs. Kyoto.doi:10.1007/978-3-319-48532-4.ISBN 978-3-319-48532-4.
Křížek, Michal; Luca, Florian; Somer, Lawrence (2001).17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics. Vol. 9. New York: Springer-Verlag.doi:10.1007/978-0-387-21850-2.ISBN 978-0-387-95332-8.MR 1866957.
Lardner, Dionysius (1840).A Treatise on Geometry and Its Application in the Arts. London: The Cabinet Cyclopædia.
Lee, Hojoo (2001)."Another proof of the Erdős–Mordell Theorem"(PDF).Forum Geometricorum.1:7–8. Archived fromthe original(PDF) on 2023-06-16. Retrieved2012-05-02.
McMullin, Daniel; Parkinson, Albert Charles (1936).An Introduction to Engineering Mathematics. Vol. 1.Cambridge University Press.
Melissen, J. B. M.; Schuur, P. C. (1995)."Packing 16, 17 or 18 circles in an equilateral triangle".Discrete Mathematics.145 (1–3):333–342.doi:10.1016/0012-365X(95)90139-C.MR 1356610.
Owen, Byer; Felix, Lazebnik;Deirdre, Smeltzer (2010).Methods for Euclidean Geometry. Classroom Resource Materials. Vol. 37. Washington, D.C.:Mathematical Association of America.ISBN 9780883857632.
Pelkonen, Eeva-Liisa; Albrecht, Donald, eds. (2006).Eero Saarinen: Shaping the Future.Yale University Press. pp. 160, 224, 226.ISBN 978-0972488129.
Petrucci, R. H.; Harwood, W. S.; Herring, F. G. (2002).General Chemistry: Principles and Modern Applications (8th ed.).Prentice Hall.ISBN 978-0-13-014329-7.
Posamentier, Alfred S.; Salkind, Charles T. (1996).Challenging Problems in Geometry.Dover Publications.
Riley, Michael W.; Cochran, David J.; Ballard, John L. (December 1982). "An Investigation of Preferred Shapes for Warning Labels".Human Factors: The Journal of the Human Factors and Ergonomics Society.24 (6):737–742.doi:10.1177/001872088202400610.
Stahl, Saul (2003).Geometry from Euclid to Knots. Prentice-Hall.ISBN 0-13-032927-4.
Svrtan, Dragutin; Veljan, Darko (2012)."Non-Euclidean versions of some classical triangle inequalities".Forum Geometricorum.12:197–209.
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White, Steven F.; Calderón, Esthela (2008).Culture and Customs of Nicaragua. Greenwood Press.ISBN 978-0313339943.
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External links

Weisstein, Eric W."Equilateral Triangle".MathWorld.

v t e Fundamental convexregular anduniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) /D_n	E₆ /E₇ /E₈ /F₄ /G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron •Cube	Demicube		Dodecahedron •Icosahedron
Uniform polychoron	Pentachoron	16-cell •Tesseract	Demitesseract	24-cell	120-cell •600-cell
Uniform 5-polytope	5-simplex	5-orthoplex •5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex •6-cube	6-demicube	1₂₂ •2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex •7-cube	7-demicube	1₃₂ •2₃₁ •3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex •8-cube	8-demicube	1₄₂ •2₄₁ •4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex •9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex •10-cube	10-demicube
Uniformn-polytope	n-simplex	n-orthoplex •n-cube	n-demicube	1_k2 •2_k1 •k₂₁	n-pentagonal polytope
Topics:Polytope families •Regular polytope •List of regular polytopes and compounds •Polytope operations

v
t
e

Polygons (List)

By number
of sides

1–10 sides	Monogon (1) Digon (2) Triangle (3) Quadrilateral (4) Pentagon (5) Hexagon (6) Heptagon (7) Octagon (8) Nonagon/Enneagon (9) Decagon (10)
11–20 sides	Hendecagon (11) Dodecagon (12) Tridecagon (13) Tetradecagon (14) Pentadecagon (15) Hexadecagon (16) Heptadecagon (17) Octadecagon (18) Icosagon (20)
>20 sides	Icositrigon (23) Icositetragon (24) Triacontagon (30) 257-gon Chiliagon (1000) Myriagon (10,000) 65537-gon Megagon (1,000,000) Apeirogon (∞)

Classes

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