Circle-less Area Puzzle - Answer
First, visualize the area as it might have been constructed:
Next, we find the area by first turning the above circles into a decagon:
First we need to find the area of a decagon with sides of 2. From geometry, we know that the
area of a decagon is:
©2011, Leon Hostetler, www.leonhostetler.com
Circle-less Area Puzzle - Answer
This works out to: 30.77684
5
𝐴 = 𝑠 2 �5 + 2√5
2
Now we need to find out the area of each of the circles. From geometry, we know that the area
of a circle is:
This works out to: 𝜋
𝐴 = 𝜋𝑟 2
Now we need to find out the portion of each small circle that is within the decagon that we drew.
We know that each vertex of the decagon lies on the center of each circle. From geometry, we
2
5
know that the interior angles of a decagon are 144°. This means that of each circle, is within
the decagon.
Now to find the area of the gray shape, we only need to subtract the area of the circles that lie
within the decagon, from the area of the decagon. So:
𝐴 = 30.77684 − 4𝜋 = 18.2105
And our answer is: 18.2105
©2011, Leon Hostetler, www.leonhostetler.com