# Reuleaux and its relation to triangles

Now, it’s not pronounced like Rolex, but more like rulers. I don’t know why, but it is.

What are Reuleaux triangles?

Think of two circles overlapping with each other, with the centre of one circle being on the other circle. Now add another circle with one of the overlaps being the new circle’s side. Did I mention that the circles have to be equal in size?

When you’re done, it will be a Venn Diagram. In the middle, there will be a triangle-like shape. That is the Reuleaux triangle.

The perimeter is easy to find: it is just half the circumference of the circle it originated from. Why? Because equilaterals have a 60 degree angle, which is 1/6th of a full rotation. So three of those 1/6th arcs make 3/6 of the original circumference, or half. So the perimeter is just pi*d.

The area. Yay…

Call the radius 1. With each of the bumps on the outside of the equilateral triangle and the equilateral triangle itself, you can make three slices of the original circle, all having the area of π/6. Now, there exists a formula for the area of an equilateral triangle given just the side. It is:

A={(√3)/4} * x^2

So the area of the slices is π/2, and the equilateral triangle is (√3)/4.Since there are three slices and there is only one equilateral triangle, we can get rid of (√3)/2. So the area of a Reuleaux triangle with side/radius of 1 is π/2 – (√3)/2, or about 0.70477

Where is this actually used?

In Bermuda, there exists coins that are shaped this way.

Also, the Reuleaux triangle was based in a film machine for the Soviet Luch-2 8mm (what?) film projector.

Interesting fact: It can rotate around a square, fitting perfectly if it is the right size of square. It remains unknown, however, how densely a Reuleaux triangle can be packed inside a plane is, but it is conjectured to be around 0.922888

Also, taking this to a whole new dimension, there exists Reuleaux tetrahedrons, which are made from four Reuleaux triangles.

There also exists Reuleaux polygons, which look so satisfying. I guess they could also be good approximations of a circle.

Okay, that’s as far as I know.

I’m sorry, to the few viewers who actually read these posts, that I couldn’t upload more frequently. School has made my day far busier, and I’ll try to upload more.

Until then, bye!

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