Math Puzzle for Quarantining Humans: 6 Inscribed Circles

Robert M. Vunabandi
7 min readJul 20, 2020

This post is part of a series of math puzzles. Here is the list of puzzles for reference:

Now, let’s continue with 6 Inscribed Circles:

6 Inscribed Circles: PROBLEM

6 Inscribed Circles

The premise of this problem is very simple. Essentially, the outer circle has radius 1, and it has 6 circles with equal radius inscribed inside it in such a way that they both touch the outer circle and touch each other. Now, the question is, if we were to put a final, 7th circle in the center of the outer circle in such a way that this 7th circle touches all the other 6 circles, what would be radius of this 7th circle?

Note that you could, if you wanted, try to guess the answer. However, for this question you must prove beyond doubt that this circle has the radius that you give. If you get stuck, scroll down to see a hint (don’t scroll too fast or else you’ll get to the solution).

6 Inscribed Circles: HINT

Here’s a hint for you. The 7th circle has the same radius as the 6 inscribed circles. Remember, you still have to prove whatever radius you found. The only thing you know are that outer circle has radius 1, and all 6 inner circles touch each other and touch the outer circles (but they do not overlap).

6 Inscribed Circles: SOLUTION

6 Inscribed Circles with the center circle drawn


Before we get to the solution, we must make ourselves aware of a few formulas and rules. These are going to be our tools to…

Robert M. Vunabandi

Learning through life experiences and books, I share my ever-evolving understanding of the world and the niche-sphere of life that I live in.