Clown hats
Weekly Problem 51 - 2015
Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?
Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?
Age
14 to 16
Challenge level
Problem
Charlie is making clown hats from a circular piece of cardboard.
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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 Charlie can make from the piece of cardboard?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Let the radius of the circular piece of cardboard be $r$. The diagram shows a sector of the circle which would make one hat, with the minor arc shown becoming the circumference of the base of the hat.
Image
The circumference of the circle is $2\pi r$. Since $$6r<2\pi r<7r$$ we can cut out $6$ hats in this fashion.
Moreover, the area of cardboard unused in cutting out any $6$ hats is less than the area of a single hat. Hence there is no possibility that more than $6$ hats could be cut out.