Area inside a circle

Geometry Level 2

Four identical circles are tangent to each other and are tangent to a bigger circle as shown. If the radius of each small circle is $r$ , what is the area of the yellow region?

r^2(2\sqrt{2}\pi -2)

2\sqrt{2}\pi r^2-4r^2

2\sqrt{2}\pi r^2 + \pi r^2

3\pi r^2 +2\sqrt{2}\pi r^2

1 solution

Marta Reece
Nov 17, 2017

Radius of the largest circle $R=r(\sqrt2+1)$

Area of the largest circle $A_1=\pi r^2 (1+\sqrt2 )^2=\pi r^2(3+2\sqrt2)$

Area of the four white circles $A_2=4\pi r^2$

Area of the white star in the middle (as a square minus four quarter-circles) $A_3=(2r)^2-4\times\frac14 \pi r^2$

Yellow area $A=A_1-A_2-A_3= \boxed{2\sqrt2\pi r^2-4r^2}$

Area inside a circle

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