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Anupam

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· 1 Months ago

question (Question:97945.0)

figure in the question is different from the one provided in the answer

Question:

In the figure shown, ABCD is a rectangle. The circles shown are equal. Find the ratio of the area of the shaded region to the area of ABCD.

Options:
A) \((4 - π) : 8\)
B) \((4 - π) : 12\)
C) \((4 – π) : 16\)
D) \((4 - π) : 24\)
Solution:
Ans: (c)

Let the radius of each circle be r units.

AB = AD = 4r units.

∴ ABCD is a square of side 4r units.

Let E, F, G and H be the centers of the circles as shown below.


EF = FG = GF = EH = 2r units.

∴ EFGH is a square of side 2r units.

Area of the region between the circles = Area of EFGH – (Area of the sectors e, f, g and h).

Area of each of the sectors e, f, g and h = \(\cfrac{90°}{360°}πr^2\) sq units.
∴ Area of 4 sectors = \(πr^2\) sq. units.
Area of the region between the circles 

\(= [(2r)^2 – πr^2]\) sq units = \((4 – π) r^2\) sq. units

\(= (4 – π) : 16.\)

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