Two semi-circles are drawn on the diameter of a semi-circle with radius $18 c m$ . A circle with centre C is drawn such that it touches the two semi-circles. Find area of shaded region.

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Solution

Draw $R M ⊥ A B$

$A M = M B = \frac{A B}{2} = 18 c m$

$A P = P M = M Q = Q B = \frac{18}{2} = 9 c m$

$M R = A M = 18 ㎝$

$C M = R M - C R = 18 - r$

$P C = P E + E C = 9 + r$

In triangle $C M P$

${P C}^{2} = {C M}^{2} + {P M}^{2} \Rightarrow {(9 + r)}^{2} = {(18 - r)}^{2} + 9^{2}$

$\Rightarrow 81 + 18 r + r^{2} = 324 - 36 r + r^{2} + 81 \Rightarrow 54 r = 324 \Rightarrow r = 6 c m$

$∴$ Area of shaded region=area of semi circle AB $- 2$ area of semi circle-area of circle with C as centre

$= \frac{1}{2} π \times 18^{2} - 2 \times \frac{1}{2} \times π \times 9^{2} - π \times 6^{2}$

$= 162 π - 81 π - 36 π = 45 π c m^{2}$

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Two semi-circles are drawn on the diameter of a semi-circle with radius 18cm. A circle with centre C is drawn such that it touches the two semi-circles. Find area of shaded region.

Two semi-circles are drawn on the diameter of a semi-circle with radius $18 c m$ . A circle with centre C is drawn such that it touches the two semi-circles. Find area of shaded region.