A geometry problem by Jose Sacramento

Geometry Level 4

The figure shows four circles, and the largest circle has a radius of 6. The other three circles are congruent and tangent to each other and to the largest circle.

What is the radius of the smallest circles? Give your answer to 3 decimal places.


The answer is 2.784.

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Jose Sacramento by Jose Sacramento , 66, Portugal

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3 solutions

h = ( 2 r ) 2 + r 2 = 4 r 2 + r 2 = r 3 h=\sqrt{(2r)^2+r^2}=\sqrt{4r^2+r^2}=r\sqrt{3}

a = 2 3 h = 2 3 ( r 3 ) = 2 r 3 3 a=\dfrac{2}{3}h=\dfrac{2}{3}(r\sqrt{3})=\dfrac{2r}{3}\sqrt{3}

Thus,

6 = 2 r 3 3 + r 6=\dfrac{2r}{3}\sqrt{3}+r

r ≈ 2.784609691 r\approx 2.784609691

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Ajit Athle
Nov 24, 2016

We may use the generalized formula: r = - abc/(ab+bc+ca-2√(abc(a+b+c))). In this case, a=b=c=x while r=6 cm which gives us: 6 = - x/(3-2√3)) or x ~ 2.27846 cm

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