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March 22, 2020

Challenge Problem 14: Cube Inside Hemisphere Puzzle

A hemisphere has a radius of 10. A cube is constructed so its bottom face has the same center as the hemisphere, and the four corners of its top face just touch the hemisphere. A green line is constructed from the center of the cube’s top face to a corner of its bottom face, as shown. What is the length of the green line in the following figure?

This is adapted from a design exam in India (UCEED exam in 2017). 

There’s a neat way to solve this without having to do any calculations! It’s based on symmetry.

The green line goes from the center of the top face to a corner of the bottom face. This is the same distance as the center of the bottom face to a corner of the top face, due to symmetry of a cube. In other words, the green and blue lines have the same length:

But notice the blue line is a radius of the hemisphere, so it’s length is 10 m. Hence the green line also has a length of 10 m. It’s simple if you think inside the box!

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