The diagram shows a square of side 4 cm with four identical semi-circles drawn with their centres at the mid-points of the sides.
The four semi-circles each touch two other semi-circles, as shown.
What is the shaded area, in cm2?
A. 8 − π B. π C. π − 2 D. π −√2 E. 8 − 2π
E. 8 − 2π
We let the vertices of the square be P, Q, R and S, and the midpoints of the edges be K, L, M and N, as shown.
KLMN is a square. You are asked to prove this in Problem 21.2.The square PQRS has sides of length 4 cm. Hence KQ = QL =2 cm.
Therefore, by Pythagoras’ Theorem, the length of KL is √(22 + 22) cm = 2√2 cm.
The part of each semicircle that is inside the square KLMN is a quarter circle.
The shaded area is the area of the square KLMN minus the areas of these four quarter circles.
The area of the square KLMN is KL2, that is, (2√2)2 cm2 = 8 cm2.
The length, 2√2 cm, of KL is the the sum of the lengths of two radii of the semicircles.
It follows that the radius of the semicircles is √2 cm.
The four quarter circles make up a circle of radius √2 cm and hence area π√22 cm2 = 2π cm2.
Therefore the shaded area is 8 cm2 − 2π cm2 = (8 − 2π) cm2.
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