Thread: More circles, arcs and sectors
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Old November 30th, 2008, 01:43 PM
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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}}
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