Semicircle

A semicircle with a radius . ${\displaystyle r}$

The semicircle describes the one-dimensional number of points that form half of a circle . The interior angle of a semicircle measures 180 ° or radian , so the semicircle is only symmetrical along one axis . Half of a circular disk is also known as a semicircle, but it is a two-dimensional shape that also includes the diameter of the circle and all of the included points. According to Thales' theorem , every triangle with two corners at the end points of a semicircle and the third corner at any position on the semicircle is a right-angled triangle with a right angle at the third corner point. All straight lines that orthogonally intersect a semicircle are copunctal . ${\displaystyle \pi }$

Use

A semicircle with arithmetic and geometric mean of the lengths and . ${\displaystyle a}$ ${\displaystyle b}$

When constructing with compasses and ruler , the semicircle can be used to derive the arithmetic and geometric mean of two lengths. In a semicircle with the diameter , the arithmetic mean of and results as the radius . If you choose the diameter again and construct an orthogonal at the point where and meet, the geometric mean results as the length from this point to the point of intersection with the semicircle. This property can be proven with the Pythagorean Theorem and can also be used to quadrature (determine the area) of a rectangle . A rectangle with the side lengths and and a square with the side length of the geometric mean and have the same area . For any shapes ( except for the circle ) for which a rectangle of the same area can be constructed, their area can also be determined. ${\displaystyle d=a+b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle r}$ ${\displaystyle d=a+b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$

Parameterization

The semicircle with radius and center that is completely above can be described by the following equation: ${\displaystyle r}$ ${\displaystyle (x_{0},y_{0})}$ ${\displaystyle y=y_{0}}$

${\displaystyle y=y_{0}+{\sqrt {r^{2}-(x-x_{0})^{2}}}}$.

The corresponding semicircle, which lies completely below , can be expressed as: ${\displaystyle y=y_{0}}$

${\displaystyle y=y_{0}-{\sqrt {r^{2}-(x-x_{0})^{2}}}}$.

Use in geometry

A salinon (blue region)

An Arbelos (gray region)

Geometric figures from Archimedes ' Book of Lemmas are often based on circular and semicircular constructions: The Salinon , a mirror-symmetrical geometrical figure, consists of four semicircles. An arbelos describes the region of a surface that is enclosed by three semicircles, which all lie on the same side of a straight line and are only connected at their corner points.

See also

• amphitheater
• Archimedean Circle
• Twin circles of Archimedes
• Salinon
• Arbelos

Web links

Wiktionary: semicircle  - explanations of meanings, word origins, synonyms, translations

• Semicircle - Mathworld

Individual evidence

1. ^ Euclid's Elements, Book VI, Proposition 13