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Explanation:

Let radius of the semicircle be R and radius of the circle be r.
Let P be the centre of semicircle and Q be the centre of the circle.
Draw QS parallel to BC.
Now, ΔPQS ~ ΔPBC

PQPB=QSBCR+r2R=R-rRR+r=2R-2rr(1+2)=R(2-1)

r=R(2-1)(2+1)×(2-1)(2-1)

r=R(2-1)2

Required Ratio = πr2πR2×2

=πR2(2-1)4×2πR2

=2(2-1)4:1

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