What is the angle subtended at the centre of

Question:

What is the angle subtended at the centre of a circle of radius 6 cm by an arc of length 3 π cm?

Solution:

We have

$r=6 \mathrm{~cm}$

length of the $\operatorname{arc}=3 \pi \mathrm{cm}$

We will find the angle subtended at the centre of a circle.

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

Substituting the values we get,

$3 \pi=\frac{\theta}{360} \times 2 \pi \times 6$....(1)

Now we will simplify the equation (1) as below,

$3 \pi=\frac{\theta}{360} \times 12 \pi$

$3 \pi=\frac{\theta}{30} \times \pi$

$3=\frac{\theta}{30}$

$\Rightarrow \quad \theta=90^{\circ}$

Therefore, angle subtended at the centre of the circle is $90^{\circ}$.