# 07 Area inside the larger circle but outside the smaller circle

**Problem**

From the figure shown below, DE is the diameter of circle A and BC is the radius of circle B. If DE = 60 cm and AC = 10 cm, find the area of the shaded region.

**Solution**

$(R - 10)^2 + 30^2 = R^2$

$(R^2 - 20R + 100) + 900 = R^2$

$20R = 1000$

$R = 50 \, \text{ cm}$

$\sin (\frac{1}{2}\theta) = \dfrac{30}{R} = \dfrac{30}{50}$

$\theta = 73.74^\circ$

Area of segment CDE

$A_{CDE} = A_{BDCE} - A_{BDE}$

$A_{CDE} = \dfrac{\pi R^2 \theta_{deg}}{360^\circ} - \frac{1}{2}R^2 \sin \theta$

$A_{CDE} = \dfrac{\pi (50^2)(73.74^\circ)}{360^\circ} - \frac{1}{2}(50^2) \sin 73.74^\circ$

$A_{CDE} = 408.76 \, \text{ cm}^2$

Required area = Area of cirlce of radius R - Area of semi-circle of radius 30 cm - Area of segment CDE

$A = \pi R^2 - \frac{1}{2}\pi (30^2) - A_{CDE}$

$A = \pi (50^2) - \frac{1}{2}\pi (30^2) - 408.76$

$A = 4617.79 \, \text{ cm}^2$ *answer*