ratio of red area to blue area

Geometry Level 1

Four small circles of equal radii are tangent to each other and their diameters lie on the diameter of a big circle as shown. What is the ratio of the red area to the blue area?

3 : 1 3:1 4 : 1.5 4:1.5 5 : 2 5:2 2 : 1 2:1

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1 solution

Let r r be the radius of one small circle, so the diameter is 2 r 2r . The radius of the big circle is 4 r 4r .

b l u e a r e a = 4 ( π ) ( r 2 ) = 4 π r 2 blue~area=4(\pi)(r^2)=4\pi r^2

r e d a r e a = a r e a o f b i g c i r c l e − b l u e a r e a = π ( 4 r ) 2 − 4 π r 2 = 16 π r 2 − 4 π r 2 = 12 π r 2 red~area=area~of~big~circle-blue~area=\pi(4r)^2-4\pi r^2=16\pi r^2 -4\pi r^2=12\pi r^2

r a t i o = 12 π r 2 4 π r 2 = 3 1 = 3 : 1 ratio=\dfrac{12\pi r^2}{4\pi r^2}=\dfrac{3}{1}=\boxed{3:1}

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