Question

Derive the formula used to find the area of sector and length of arc of sector. [2 MARKS]

Answer 1

Concept: 1 Mark each

Circle is also a sector subtending an angle of ${360}^{\circ }$ at the centre.

So, when the degree measure of the angle at the centre is 1, area of the sector
$=\frac{\pi r^2}{360}$

Therefore, when the degree measure of the angle at the centre is $\theta$,

Area of the sector
$=\frac{\pi r^2}{360}\times \theta =\frac{\theta }{360}\times \pi r^2$

Arc of a sector is a part of the circumference.

When the degree measure of the angle at the centre is 1, the length of the arc
$=\frac{2\pi r}{360}$

Therefore, when the degree measure of the angle at the centre is $\theta$,

Length of the arc
$=\frac{2\pi r}{360}\times \theta =\frac{\theta }{360}\times 2\pi r$