Why a triangle can roll like a circle

2026-05-13

Roll a circle between two rulers and the rulers never separate. The rulers never gouge. The gap between them is just the diameter, and the circle does its job of being the same width no matter which way you turn it.

Nearly everyone assumes circles are the only shape that pulls this off. They aren't.

The curved triangle above is a Reuleaux triangle. Three corners, three arcs, definitely not a circle. The dashed grey circle behind it is a genuine circle of the same width, for comparison. Drag the slider and watch the triangle spin between the rails. The gap on the right - 1.000, always - is the punchline.

How to build one

Start with an equilateral triangle of side s. Put the point of a compass on one corner and set the radius to s - exactly the side length. Sweep an arc from one of the other corners to the third. Do this from all three corners. The three arcs meet at the corners and bulge outward into a soft, cornered, almost-circle.

Every point on every arc sits at distance s from the opposite corner. That's the whole construction. Three points, one compass, one radius.

Why it has the property, in one line

Pin the triangle between two parallel rails so it touches both. Wherever it touches a rail, the contact is either a corner or a smooth bit of an arc.

Suppose a corner is on the top rail. Then the bottom rail touches the arc opposite that corner. And here is the trick: that arc is, by construction, a circle of radius s centred on the corner that's on top. So the bottom rail is exactly s below the top one.

Suppose instead it's the other way around - the bottom rail touches a corner and the top rail rests on an arc. Same trick. The arc is a circle of radius s centred on the corner on the bottom, so the top rail is s above it.

gap between rails = one radius of one arc = side length s

That's it. The dashed line inside the figure that the slider drags around is exactly that radius. Watch it - it never gets longer, never gets shorter. The proof has been hiding in plain sight the whole time.

Things that follow

Manhole covers in some cities are not round - they are Reuleaux. A Reuleaux cover can't fall through its own hole no matter how you twist it, for the same reason a circle can't: same width every way. Tucked inside a special chuck, a Reuleaux-shaped drill bit will trace out an almost-square hole - close enough that machinists do use it. Currencies have minted Reuleaux coins so that vending machines can size them with a single slot, the same way they size a round one.

Three arcs, one compass radius, a small theorem - and an entire class of shapes hiding behind the circle the whole time.

Stitch three circular arcs to the corners of an equilateral triangle and the shape that pops out has constant width: the same diameter in every direction. The two parallel rails always meet a corner on one side and the arc centred on it on the other. The gap is just one radius.