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Seven small circles of radius 2 are cut from the large circl

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by Max@Math Revolution » Thu Dec 13, 2018 11:45 pm

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[Math Revolution GMAT math practice question]

Seven small circles of radius 2 are cut from the large circle, as shown. The small circles are tangent each other, and all small circles except for the center circle are tangent to the larger circle. What is the area of the shaded region?

Image

A. 8 π
B. 20 π
C. 32 π
D. 48 π
E. 64 π
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by fskilnik@GMATH » Fri Dec 14, 2018 1:49 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

Seven small circles of radius 2 are cut from the large circle, as shown. The small circles are tangent each other, and all small circles except for the center circle are tangent to the larger circle. What is the area of the shaded region?

Image

A. 8 π
B. 20 π
C. 32 π
D. 48 π
E. 64 π
Image

$$? = \pi \cdot {6^2} - 7 \cdot \pi \cdot {2^2} = 8\pi $$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Sun Dec 16, 2018 5:26 pm
=>

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The radius of the large circle is 2*2 + 2 = 6. Thus, the area of the large circle is 6^2Ï€ = 36Ï€, and the area of each of the seven small circles is 2^2Ï€ = 4Ï€.
Thus, the area of the shaded region is 36Ï€ - 7(4Ï€) = 8Ï€.
Therefore, the answer is A.
Answer: A
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