More circles, arcs and sectors

loizoud94 · #1 $Old$ November 30th, 2008, 02:21 PM

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

skeeter · #2 $Old$ November 30th, 2008, 02:42 PM

$s = r\theta$

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

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

Soroban · #3 $Old$ November 30th, 2008, 02:43 PM

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

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