You are making clown hats from a circular piece of cardboard.

The circumference of the base of each hat equals its slant height,
which in turn is equal to the radius of the piece of cardboard.

What is the maximum number of hats that you can make from
the piece of cardboard?

As usual, I encourage to pause the article, grab your pen and paper, and give this a go. When you are ready, keep reading for the solution.

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Solution

Let’s say the radius of the circle is r, then the slant height of each hat must also be r.

In the figure above, we have a sector of the circle in which the circular part of the boundary also has length r. And that’s the base of the hat we are making!

We also know that the circumference of a circle with radius r is 2πr.

We know π is just a bit greater than 3, so

And if we multiply this inequality by 2r, we get

This means the circumference of the circle can make just a little less than 7 hats.

So we will get 6 clown hats 🤡🤡🤡🤡🤡🤡

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Extension Problems from UKMT

Let me know how you get on with these in the comments!