Math Puzzle for Quarantining Humans: 6 Inscribed Circles

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

• 3333
• 6 Inscribed Circles (this post)
• Slicing the Parabola
• 4444
• […coming soon…]
• […coming soon…]

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).

Photo by Markus Winkler on Unsplash

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).

Photo by Markus Winkler on Unsplash

6 Inscribed Circles: SOLUTION

6 Inscribed Circles with the center circle drawn

Tools

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…