Hello, loizoud94!

The length of an arc of a circle is 12 cm.

The corresponding sector area is 108 cm².

Find:

a) the radius of the circle.

b) the angle subtended at the centre of the circle by the arc. Code:

* * * A
* *
* / *
* r / *
/
* / θ *
* * - - - - * B
O r

Length of arc: .$\displaystyle s \:=\:r\theta$

. . We have: .$\displaystyle r\theta \:=\:12$ .[1]

Area of sector: .$\displaystyle A \:=\:\tfrac{1}{2}r^2\theta$

. . We have: .$\displaystyle \tfrac{1}{2}r^2\theta \:=\:108 \quad\Rightarrow\quad r^2\theta \:=\:216$ .[2]

Divide [2] by [1]: .$\displaystyle \frac{r^2\theta}{r\theta} \:=\:\frac{216}{12} \quad\Rightarrow\quad\boxed{ r \:=\:18\text{ cm}}$

Substitute into [1]: .$\displaystyle 18\theta \:=\:12 \quad\Rightarrow\quad \boxed{\theta \:=\:\frac{2}{3}\text{ radians}}$