Question

Find the area of the colored region in the figure if the radius of the circle is 7 cm.

Answer 1

So we can find the area of the segment by subtracting the area of the triangle from the area of the sector ABC.

i.e. Area of the segment = Area of the sector AOB – �� Area of the triangle AOB

Area of the sector =$\frac{120}{360}\times \pi \times 7^2$

= $\frac{1}{3}\times \frac{22}{7}\times 7^2$

= 51.33$c{m}^2$

Now let us consider the $\Delta$ ��AOB,

We know that the perpendicular bisector of a chord will pass through the Centre of the circle. Assume the perpendicular bisector intersects AB at C.

Then in rt triangle $(\frac{r}{\sqrt{2}}{)}^2$ $\frac{120}{360}\times \pi \times 7^2$

$△��ACO,\angle ACO={90}^{\circ },\angle AOC={60}^{\circ }$(AC bisects the angle $\angle AOB$ and AO = 7 cm.

Applying trigonometric ratios,

OC = AO cos(60) = 7 cos(60) = 3.5 cm

AB = AO sin (60) = 7 sin(60) = 6.06 cm

Area of triangle ��AOB = $\frac{1}{2}\times AB\times OC$

=$\frac{1}{2}\times (2\times 6.06)\times 3.5$

= 21.21 $c{m}^2$

Therefore the area of the colored area = 51.33 – 21.21 = 30.12 $c{m}^2$