Calculations at a Reuleaux triangle. This is an equilateral triangle with the side length a, where around each vertex a circle with the radius a is drawn. The three resulting circular segments are added to the equilateral triangle. The Reuleaux triangle is also the intersection of those three circles. It is the simplest form of a curve of constant width, this width is also a. Its two-dimensional equivalent is the Reuleaux tetrahedron.
Enter one value and choose the number of decimal places. Then click Calculate.
Formulas:
l = a π / 3
p = a π
A = ( π - √3 ) a² / 2
pi:
π = 3.141592653589793...
Radius, length and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
A shape that has the same thickness or width everywhere is called a curve of constant width. If you put this shape between two parallel lines so that they touch the curve of constant width, you can rotate it within these lines so that the contact is maintained. There is never a gap between the shape and the lines or the shape extends beyond these lines. The center of the curve of constant width does not have to be exactly in the middle of the lines and can change its position by rotating it.
The only curve of constant width with a fixed center is the circle, which is the trivial case of a curve of constant width and the one with the largest area per width. The one with the smallest area per width is the Reuleaux triangle. In between are the regular Reuleaux polygons, which get closer and closer to a circle as the number of corners increases. A Reuleaux polygon is only for an odd number of corners a curve of constant width.
The Reuleaux triangle was already known before Franz Reuleaux, and Leonardo da Vinci used it to map the earth. Franz Reuleaux researched in the field of gear theory and used this form for mechanical applications, therefore this form was named after him.