Cube in a Cone, Version 1.1

Geometry Level 3

A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that four of its vertices are located in the base of the cone and the remaining four in the sides of the cone. Find the side length of the cube.

If you get your answer in the format a b − c d \frac{a\sqrt b-c}d report a + b + c + d a+b+c+d


The answer is 24.

Marta Reece by Marta Reece , 69, USA

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1 solution

Tom Engelsman
Oct 1, 2020

If the cube has side length s s , then let us take a cutting plane through the cone along the cube's diagonal (of length 2 s \sqrt{2}s ) so that this plane is normal to the cube's top face. This results in the side proportion of two similar right triangles:

( s / 2 ) 2 3 − s = 1 3 ⇒ s = 9 2 − 6 7 \frac{(s/2)\sqrt{2}}{3-s} = \frac{1}{3} \Rightarrow \boxed{s = \frac{9\sqrt{2}-6}{7}}

which a + b + c + d = 24. a+b+c+d = 24.

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