# More circles, arcs and sectors

• Nov 30th 2008, 12:21 PM
loizoud94
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
• Nov 30th 2008, 12:42 PM
skeeter
$\displaystyle s = r\theta$

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

you are given $\displaystyle s$ and $\displaystyle A$ ... see the substitution you need to do?
• Nov 30th 2008, 12:43 PM
Soroban
Hello, loizoud94!

Quote:

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