Circle Covering

Geometry Level 3

You need at least three circles to completely cover a circle of a slightly larger radius.

The 3 overlapping orange circles have the same radius as the black circle. It takes 3 of these circles to cover the slightly larger dashed circle. The 3 overlapping orange circles have the same radius as the black circle. It takes 3 of these circles to cover the slightly larger dashed circle.

What is the smallest radius you can choose for the smaller circles to cover a unit circle?

(Round to three decimal places)


The answer is 0.866.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Otto Bretscher
Dec 19, 2018

A circle of radius r r can cover an arc of at most θ = 2 arcsin ⁡ r \theta=2\arcsin r of the unit circle (since the endpoints of the arc will have a distance ≤ 2 r \leq 2r ). To get the job done with three circles, we need θ = 2 arcsin ⁡ r ≥ 2 π 3 \theta=2\arcsin r\geq \frac{2\pi}{3} or r ≥ 3 2 r \geq \frac{\sqrt{3}}{2} . It is easy to see that the radius r = 3 2 ≈ 0.866 r=\frac{\sqrt{3}}{2}\approx \boxed{0.866} is attained.

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