Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.7 AMC 10/12 Find the answer United States

Problem:
Compute the surface area of a cube inscribed in a sphere of surface area $\pi$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:
The sphere's radius $r$ satisfies $4 \pi r^{2} = \pi \Rightarrow r = 1/2$ , so the cube has body diagonal $1$ , hence side length $1/\sqrt{3}$ . So, its surface area is $6(1/\sqrt{3})^{2} = 2$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.