Five circles in a square

Geometry Level 3

Given that the side length of the square above is 4 units and all circles are identical, what is the area of the gray region?

4 π − 4 4\pi-4 π − 1 \pi-1 2 π − 4 2\pi - 4 4 − π 4-\pi

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Consider my diagram. The area we want to know is equal to the area of circle with diameter 2 units minus the area of the blue region. Note that the area of the blue region is equal to the area of the red region. The area of the red region is equal to the area of the square with side length 2 units minus the area of the circle. We have

a r e a o f r e d r e g i o n = a r e a o f b l u e r e g i o n = 2 2 − π 4 ( 2 2 ) = 4 − π area~of~red~region=area~of~blue~region=2^2-\dfrac{\pi}{4}(2^2)=4-\pi

The required area is

a r e a o f g r a y r e g i o n = π 4 ( 2 2 ) − ( 4 − π ) = π − 4 + π = 2 π − 4 area~of~gray~region=\dfrac{\pi}{4}(2^2)-(4-\pi)=\pi-4+\pi=\boxed{2\pi-4}

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