Find the angle subtended at the centre of a circle of radius 5 cm by an arc of length ($\frac{5\pi}{3}$) cm.

Given:

Radius of the circle $=5\ cm$

Length of the arc $=\frac{5\pi}{3}\ cm$

To do:

We have to find the angle subtended at the centre.

Solution:

Let $\theta$ be the angle subtended by the arc at the centre.

This implies,

$2 \pi r(\frac{\theta}{360^{\circ}})=\frac{5 \pi}{3}$

$\Rightarrow 2 \pi \times 5 \times \frac{\theta}{360^{\circ}}=\frac{5 \pi}{3}$

$\Rightarrow \frac{\theta}{360^{\circ}}=\frac{5 \pi}{3} \times \frac{1}{10 \pi}$

$\Rightarrow \frac{\theta}{360^{\circ}}=\frac{1}{6}$

$\Rightarrow \theta=\frac{360^{\circ}}{6}$

$\Rightarrow \theta=60^{\circ}$

The angle subtended at the centre is $60^{\circ}$.