When Triangles Roll: The Magic of Constant Width Shapes

You’ve probably learned that a circle is the only shape with the same width all around. It’s why wheels roll smoothly. But here’s a plot twist: there are non-circular shapes—Reuleaux triangles, pentagons and more—that also have constant width. In simple terms, constant width means if you measure the distance between any two parallel lines just touching the shape, that distance is always the same. We’ll dive into these wild shapes, their unexpected origins, and how engineers and designers still use them today.

Where did this come from?

Back in the 19th century, German engineer Franz Reuleaux studied mechanical linkages—systems of rods and joints that transfer motion. He noticed you could trace an equilateral triangle with rounded edges (now called the Reuleaux triangle) and it would have the same width as you spun it. Earlier, the idea of constant-width curves dated back to French mathematicians in the 1700s, but Reuleaux turned it into practical mechanics. His work helped inventors create more efficient engines and drilling tools.

Where you’ll see this in real life

1. Drilling square holes: Reuleaux-triangle drill bits fit into special chucks to carve out nearly perfect square holes in wood or metal, since the bit’s constant width keeps it snug in the rotating frame. 2. Coin designs: Several countries mint coins with constant-width shapes (like the UK’s 20p and 50p) so vending machines can measure diameter to sort them, yet still give coins a distinctive look. 3. Bollards and manhole covers: Some safety posts (bollards) use these curves, so they always present the same width to traffic barriers. Similarly, constant-width manhole covers won’t fall through their circular frames. 4. Architecture and art: From modern facades to installation sculptures, designers exploit the pleasing curves of these shapes to create rolling doors or eye-catching patterns.

A common misconception

Many students assume only perfect circles have that “same-width” property. It’s easy to picture measuring a circle’s diameter with a ruler and getting the same number every time—but that’s just one example. Constant-width shapes must be convex (no dents) and have smooth boundaries, but they don’t need perfect circular arcs everywhere. Realizing there’s a family of these shapes is a fun twist that shows how geometry can surprise us.