Tightest sphere about three spheres

Geometry Level pending

Three spheres of radii 5 , 7 , 10 5 , 7, 10 are placed in space tangent to each other. Find the radius of the tightest sphere that surrounds them and is tangent to each one of them.


The answer is 17.0737.

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

David Vreken
Feb 27, 2021

The cross-section passing through the centers of the three spheres would be three mutually tangent circles, and the tightest sphere that surrounds them would be another mutually tangent circle circumscribing the original three:

By Descartes' Theorem , the radius r r of the large sphere satisfies the equation:

( 1 5 + 1 7 + 1 10 − 1 r ) = 2 ( 1 5 2 + 1 7 2 + 1 1 0 2 + 1 r 2 ) \bigg(\cfrac{1}{5} + \cfrac{1}{7} + \cfrac{1}{10} - \cfrac{1}{r}\bigg) = 2\bigg(\cfrac{1}{5^2} + \cfrac{1}{7^2} + \cfrac{1}{10^2} + \cfrac{1}{r^2}\bigg)

which solves to r = 2170 + 77 271 ≈ 17.0738 r = \cfrac{2170 + \sqrt{77}}{271} \approx \boxed{17.0738} .

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