# 7.  A chord of a circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. (Use $\pi = 3.14$ and $\sqrt3 = 1.73$)

D Devendra Khairwa

For the area of the segment, we need the area of sector and area of the associated triangle.

So, the area of the sector is :

$=\ \frac{120^{\circ}}{360^{\circ}}\times \pi \times 12^2$

or                                        $=\ 150.72\ cm^2$

Now, consider the triangle:-

Draw a perpendicular from the centre of the circle on the base of the triangle (let it be h).

Using geometry we can write,

$\frac{h}{r}\ =\ \cos 60^{\circ}$

or                                      $h\ =\ 6\ cm$

Similarly,                            $\frac{\frac{b}{2}}{r}\ =\ \sin 60^{\circ}$

or                                      $b\ =\ 12\sqrt{3}\ cm$

Thus the area of the triangle is :

$=\ \frac{1}{2}\times 12\sqrt{3}\times 6$

or                                         $=\ 62.28\ cm^2$

Hence the area of segment is:$=\ 150.72\ -\ 62.28\ =\ 88.44\ cm^2$.

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