Semicircle

For a surveying instrument, see Graphometer.

A semicircle of radius r.

In mathematics (more specifically geometry), a semicircle is a two-dimensional geometric shape that forms half of a circle. Being half of a circle's 360°, the arc of a semicircle always measures 180°. A triangle inscribed in a semicircle is always a right triangle.

Uses

A semicircle can be used to construct arithmetic and geometric means of two lengths using straight-edge and compass. If we make a semicircle with a diameter of a+b, then the length its radius is the arithmetic mean (since it's 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 end and the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean, which can be proved using Pythagorean theorem.

This method can be used to accomplish quadrature of a rectangle (since a square whose sides are equal to geometric mean of sides of a rectangle has the same area as the rectangle), and thus any figure for which we can construct a rectangle with equal area, such as any polygon (but not a circle).

Area and Perimeter

The area is:

$A=\frac{\pi}{2}r^2$

The perimeter is:

$P=\pi r$

Circumscribed Semicircle

The area is:

$A=\frac{\pi}{32}\left(\frac{abc}{A}\right)^2$

$A=\frac{\pi}{2}\left(\frac{a^2b^2c^2}{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}\right)$

$A=\frac{\pi}{8}s^2$

$A=\frac{\pi}{2}\left(1-\frac{2}{n}\right)^2\tan^2\left(\frac{180}{n}\right)s^6$

$A=\frac{\pi}{8}\frac{s^2}{\sin^2\left(\frac{180}{n}\right)}$

The perimeter is:

$P=\frac{\pi}{4}\frac{abc}{A}$

$P=\pi\frac{abc}{\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}$

$P=\pi\left(1-\frac{2}{n}\right)\tan\left(\frac{180}{n}\right)s^3$

$P=\frac{\pi}{2}s$

$P=\frac{\pi}{2\sin\left(\frac{180}{n}\right)}s$

Inscribed Semicircle

The area is:

$a=2\pi\left(\frac{A}{P}\right)^2$

$a=\frac{\pi}{8}\left(\frac{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}{(a+b+c)^2}\right)$

$a=\frac{\pi}{8}\left(\frac{\sin\left(\frac{360}{n}\right)}{1+\cos\left(\frac{180}{n}\right)+\sin\left(\frac{180}{n}\right)}\right)^2s^2$

$a=\frac{\pi}{72}\left(\frac{n}{(n-2)\tan\left(\frac{180}{n}\right)}\right)^2s^2$

$a=\frac{\pi}{8\tan^2\left(\frac{180}{n}\right)}s^2$

The perimeter is:

$p=2\pi\frac{A}{P}$

$p=\frac{\pi}{4}\left(\frac{\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}}{a+b+c}\right)$

$p=\pi\frac{\sin\left(\frac{360}{n}\right)}{2\left(1+\cos\left(\frac{180}{n}\right)+\sin\left(\frac{180}{n}\right)\right)}s$

$p=\pi\frac{n}{(6n-12)\tan\left(\frac{180}{n}\right)}s$

$p=\frac{\pi}{2\tan\left(\frac{180}{n}\right)}s$

See also

• Thales' theorem

External links

• Semicircle - Mathworld

ca:Semicercle pl:Półokrąg pt:Semicírculo tl:Kalahating bilog