The Reuleaux triangle

What just happened?

Intuitively it seems that only the circle has “constant width” — i.e. that the distance between two parallel tangent lines is the same regardless of direction. Not true! There is a whole family of constant-width figures, and the most famous of them is the Reuleaux triangle, named after the German engineer Franz Reuleaux (19th c.).

How is the Reuleaux triangle built?

We take an equilateral triangle with side $w$ and replace each side with an arc of a circle drawn from the opposite vertex with radius $w$. Such a figure has the same width in every direction:

where $h(\theta)$ is the distance from the centre to the tangent with normal $\theta$. The sum of widths in two opposite directions always equals $w$ — which is why the upper rail does not budge while rolling.

Why can it roll like a circle?

Thanks to its constant width an object placed on rollers with a Reuleaux cross-section moves smoothly, without bobbing — the top plate stays at the same height the whole time. The difference from a circle: the centre of the figure does not move in a straight line — the centroid of the Reuleaux triangle bobs gently (the centre-path plot shows a wave, a straight line for the circle).

Barbier’s theorem:

Every figure of constant width $w$ has the same perimeter $\pi w$ — exactly that of a circle of diameter $w$. Surprising, because the Reuleaux triangle looks “thinner” than a circle, yet the perimeter is identical.

Applications:

Reuleaux-shaped drill bits drill (almost) square holes, the shape appears in mechanisms (the Wankel engine), and even in coins and manhole covers. It is a striking example that an “obvious” geometric intuition can be wrong.

Why this matters for the exam:

It is a “wow” beyond the core curriculum (the curriculum: plane figures, the circle), but it is great training in thinking: not every “obvious” geometric property is true — it is worth looking for counterexamples before you take something for granted.

Remember: the circle is not the only constant-width figure — the Reuleaux triangle also has $w=\text{const}$ in every direction, can roll, and its perimeter is $\pi w$.