Find the area of the blue circle

Geometry Level pending

Each side of the square measures 4 4 units. Find the area of the blue circle. Give your answer to the nearest integer.

5 square units 9 square units 7 square units 3 square units

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1 solution

Let the square be A B C D ABCD and the center of the blue circle be O O . Due to symmetry, the circle touch the base of A B C D ABCD at the midpoint M M or C D CD . By Pythagorean theorem , we have:

O M 2 + D M 2 = O D 2 r 2 + 2 2 = ( 4 r ) 2 = r 2 8 r + 16 8 r = 12 r = 12 8 = 3 2 \begin{aligned} OM^2 + DM^2 & = OD^2 \\ r^2 + 2^2 & = (4-r)^2 = r^2 - 8r + 16 \\ 8r & = 12 \\ \implies r & = \frac {12}8 = \frac 32 \end{aligned}

Then the area of the blue circle is ( 3 2 ) 2 π 7 \left(\frac 32\right)^2\pi \approx \boxed 7 .

Great solution!

Mahdi Raza - 1 year, 1 month ago

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Glad that you like it.

Chew-Seong Cheong - 1 year, 1 month ago

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