# Thread: More circles, arcs and sectors

1. ## More circles, arcs and sectors

I don't usually struggle with maths, but if anybody could help me, it would be appreciated again!

The length of an arc of a circle is 12cm. The corresponding sector area is 108cm squared. Find:
a) the radius of the circle.
b) the angle subtended at the centre of the circle by the arc.

See attached and check if I have drawn the correct diagram.

Help is much appreciated again,
Thanks
D Loizou

2. $s = r\theta$

$A = \frac{r^2 \theta}{2} = \frac{r(r\theta)}{2}$

you are given $s$ and $A$ ... see the substitution you need to do?

3. 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: . $s \:=\:r\theta$

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

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

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

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

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