Question

Derive the formula used to find the area of segment of circle. [4 MARKS]

Answer 1

Concept: 1 Mark
Application: 3 Marks

Area of the segment APB = Area of the sector OAPB - Area of $\Delta OAB$

$=\frac{\theta }{360}\times \pi r^2-Areaof\Delta OAB$



Draw $OD\perp AB$

$cos\frac{\theta }{2}=\frac{OD}{r}$

$OD=rcos\frac{\theta }{2}\dots \dots \left(1\right)$

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

$AD=rsin\frac{\theta }{2}$

$AB=2AD$

$AB=2rsin\frac{\theta }{2}\dots \dots \left(2\right)$

$Areaof\Delta OAB=\frac{1}{2}\times AB\times OD$

$=\frac{1}{2}\times 2rsin\frac{\theta }{2}\times rcos\frac{\theta }{2}$

$=r^{2}sin\frac{\theta }{2}cos\frac{\theta }{2}$

Area of the segment

$=\frac{\theta }{360}\times \pi r^2-r^{2}sin\frac{\theta }{2}cos\frac{\theta }{2}$

$=r^2(\frac{\theta \pi }{360}-sin\frac{\theta }{2}cos\frac{\theta }{2})$