mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

(and more specifically

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

), a semicircle is a one-dimensional locus of points that forms half of a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

. The full arc of a semicircle always measures 180° (equivalently,

radians The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...

, or a half-turn). It has only one line of symmetry ( reflection symmetry). In non-technical usage, the term "semicircle" is sometimes used to refer to a half- disk, which is a two-dimensional geometric shape that also includes the diameter segment from one end of the arc to the other as well as all the interior points. By Thales' theorem, any

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...

in a semicircle with a

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

at each of the endpoints of the semicircle and the third vertex elsewhere on the semicircle is a right triangle, with a

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...

at the third vertex. All lines intersecting the semicircle

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...

ly are concurrent at the center of the circle containing the given semicircle.

Uses

A semicircle can be used to construct the

arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

and geometric means of two lengths using straight-edge and compass. For a semicircle with a diameter of ''a'' + ''b'', the length of its

radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...

is the arithmetic mean of ''a'' and ''b'' (since the radius is half of the diameter). The geometric mean can be found by dividing the diameter into two segments of lengths ''a'' and ''b'', 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 the Pythagorean 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 lengths ''a'' and ''b''. The construction of the geometric mean can be used to transform any rectangle into a square of the same area, a problem called the quadrature 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 a

lemma Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), ...

in a general method for transforming any polygonal shape into a similar copy of itself with the area of any other given polygonal shape.Euclid's Elements, Book VI, Proposition 25
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Equation

The equation of a semicircle with midpoint

(x_0,y_0)

on the diameter between its endpoints and which is entirely concave from below is :

y=y_0+\sqrt.

If it is entirely concave from above, the equation is :

y=y_0-\sqrt.

Arbelos

An arbelos is a region in the plane bounded by three semicircles connected at the corners, all on the same side of a straight line (the ''baseline'') that contains their

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...

References

{{reflist

External links

Semicircle - Mathworld
Elementary geometry es:Semicírculo

Uses

Equation

Arbelos

See also

References

External links