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), a semicircle is a one-dimensional locus of points that forms half of a circle. The full

arc ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...

of a semicircle always measures 180° (equivalently, radians, 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 Disk or disc may refer to: * Disk (mathematics), a geometric shape * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disk (functional analysis), a subset of a vector sp ...

, which is a two-dimensional

geometric shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on ...

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 inscribed in a semicircle with a vertex 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 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

at the third vertex. All lines intersecting the semicircle perpendicularly are concurrent at the center of the circle containing the given semicircle.

Uses

A semicircle can be used to

construct Construct, Constructs or constructs may refer to: * Construct (information technology), a collection of logic components forming an interactive agent or environment ** Language construct * ''Construct'' (album), a 2013 album by Dark Tranquillity ...

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 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 ca ...

means of two lengths using straight-edge and compass. For a semicircle with a diameter of ''a'' + ''b'', the length of its radius is the arithmetic mean of ''a'' and ''b'' (since the radius is half of the diameter). The

geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...

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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

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), a ...

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 diameters.

References

{{reflist

External links

Semicircle - Mathworld
Elementary geometry es:Semicírculo

Uses

Equation

Arbelos

See also

References

External links