Question

An arc of length 4 $\mathrm{\pi }$ cm subtends an angle of 40° at the centre. Find the radius and the area of the sector formed by this arc.

Answer 1

Given:
Length of the arc, l = 4$\mathrm{\pi }$ cm
Corresponding central angle, $\mathrm{\theta }$ = 40$°$
We know:
Length of the arc, l = $\frac{\mathrm{\theta }}{360}\times 2\mathrm{\pi }r$
$\Rightarrow$4$\mathrm{\pi }$ = $\frac{40}{360}\times 2\times \mathrm{\pi }\times r$

$\Rightarrow$r = $\frac{4\times 360}{40\times 2}$

= 18 cm
Also,
Area of a sector = $\frac{\mathrm{\theta }}{360}\times \mathrm{\pi }{r}^2$

∴ Area of the sector = $\frac{40}{360}\times \mathrm{\pi }\times {18}^2$

= 36$\mathrm{\pi }$ cm²