20. The diagram shows seven circular arcs and a heptagon with equal sides but unequal angles. The sides of the heptagon have length 4 . The centre of each arc is a vertex of the heptagon, and the ends of the arc are the midpoints of the two adjacent sides.
What is the total shaded area?
A $12 \pi$
B $14 \pi$
С $16 \pi$
D $18 \pi$
E $20 \pi$
20. D First note that as the length of each side of the heptagon is 4, the radius of each of the seven arcs is 2 . The sum of the interior angles of a heptagon is $(7-2) \times 180^{\circ}=900^{\circ}$. So the sum of the angles subtended by the circular arcs at the centres of the circles of radius $2 \mathrm{~cm}$ is $7 \times 360^{\circ}-900^{\circ}=$ $\left(7-\frac{5}{2}\right) \times 360^{\circ}=\frac{9}{2} \times 360^{\circ}$. Therefore the total shaded area is equal to the total area of $\frac{9}{2}$ circles of radius 2 . So the total shaded area is $\frac{9}{2} \times \pi \times 2^{2}=18 \pi$.