Question

How perimeter of segment of circle is $(\frac{\pi r\theta }{{108}^o}+2r\sin \frac{\theta }{2})$. Give derivation.

Answer 1

R.E.F image

Perimeter of the circle $\left(2\pi r\right)$

APBO area of their sector $=\pi r^2\times \frac{\theta }{360}$

$=\frac{\pi r^2\theta }{360}$

$sin\theta /2=\frac{AD}{r}$

$AD=r\frac{sin\theta }{2}$

Area of $△APD=\frac{1}{2}\times AB\times COD$

$=\frac{1}{2}\times 2\times AD\times OD$

$=\frac{1}{2}\times 2r\frac{sin\theta }{2}rcos\theta /2=\frac{r^2}{2}sin\theta$

area of segment $=\frac{\pi r^2\theta }{360}-\frac{2r^{2}sin\theta }{2}$