Reuleaux Triangle
What non-circle shape can be rotated inside a square and touch all four sides of the square at the same time? All shapes of constant width can do that. Here is an example. Find the perimeter of this shape (the reuleaux triangle) if the outside square's perimeter is 16.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
If you have 3 equally sized circles intersecting each other at their centers the center intersection shape is a Reulaux triangle. Notice an equilateral triangle can be inscribed which means that a side of the Reulaux triangle that belongs to the circle is a sixth of it's perimeter. The radius of the circle must be the same as the side length of the square for the Reulaux triangle to be exactly fitting. The side length of the square is 4. Therefore the radius of the circles is 4 meaning that their perimeter is 8 π . The perimeter of the Reulaux triangle is 3 sixths of the perimeter of one of the circles. Therefore the answer is 4 π ≈ 1 2 . 5 6 .