Inmathematics (and more specificallygeometry), asemicircle is a one-dimensionallocus of points that forms half of acircle. It is acircular arc that measures 180° (equivalently,πradians, or ahalf-turn). It only has one line of symmetry (reflection symmetry).
In non-technical usage, the term "semicircle" is sometimes used to refer to either aclosed curve that also includes the diameter segment from one end of the arc to the other or to the half-disk, which is a two-dimensionalgeometric region that further includes all the interior points.
ByThales' theorem, anytriangleinscribed in a semicircle with avertex at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is aright triangle, with aright angle at the third vertex.
All lines intersecting the semicircleperpendicularly areconcurrent at the center of the circle containing the given semicircle.
A semicircle can be used toconstruct thearithmetic andgeometric means of two lengths using straight-edge and compass. For a semicircle with a diameter ofa +b, the length of itsradius is the arithmetic mean ofa andb (since the radius is half of the diameter).
Thegeometric mean can be found by dividing the diameter into two segments of lengthsa andb, and then connecting their common endpoint to the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean. This can be proven by applying thePythagorean theorem to three similar right triangles, each having as vertices the point where the perpendicular touches the semicircle and two of the three endpoints of the segments of lengthsa andb.[1]
The construction of the geometric mean can be used to transform any rectangle into a square of the same area, a problem called thequadrature of a rectangle. The side length of the square is the geometric mean of the side lengths of the rectangle. More generally, it is used as alemma in a general method for transforming any polygonal shape into a similar copy of itself with the area of any other given polygonal shape.[2]
TheFarey sequence of ordern is thesequence of completely reducedfractions which whenin lowest terms havedenominators less than or equal ton, arranged in order of increasing size. With a restricted definition, each Farey sequence starts with the value 0, denoted by the fraction0/1, and ends with the fraction1/1.Ford circles can be constructedtangent to their neighbours, and to the x-axis at these points. Semicircles joining adjacent points on the x-axis pass through the points of contact at right angles.[3]
The equation of a semicircle with radius and midpoint on the diameter between its endpoints and which is entirely concave from below is
If it is entirely concave from above, the equation is
Anarbelos is a region in theplane bounded by three semicircles connected at their endpoints, all on the same side of astraight line (thebaseline) that contains theirdiameters.
