How is the perimeter formula pi times w derived?

This follows from Barbier's theorem, which states that every curve of constant width w, regardless of its exact shape, has perimeter exactly pi times w, the same as a circle of diameter w. For the Reuleaux triangle this can also be checked directly by summing the three 60-degree arcs of radius w.

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🔘 Reuleaux Polygons

Interactive Reuleaux polygon simulator. Build genuine curves of constant width from odd-sided regular polygons (3, 5, 7, 9, 11 sides), watch one rotate smoothly while inscribed in a square, and see live width, area and perimeter statistics.

Geometry2DEasy60 FPS

🔘 What It Demonstrates

A curve of constant width is a convex shape whose width — the distance between two parallel supporting lines — is the same in every direction, just like a circle, but without being a circle. Reuleaux polygons are the classic family of non-circular examples, built from a regular polygon with an odd number of sides (3, 5, 7, 9, 11, …). For each vertex of the base polygon, draw a circular arc centred at that vertex, with radius equal to the distance to the two vertices farthest from it (the "opposite" vertices) and spanning between them. The resulting curve — n arcs for an n-sided base polygon — has exactly constant width equal to the polygon's diagonal/side-defining distance, because the arc centred at any vertex always passes through points exactly that same fixed distance away.

The simplest and most famous case is the Reuleaux triangle (n = 3): start from an equilateral triangle and draw three arcs, each centred at one vertex with radius equal to the triangle's side length, connecting the other two vertices. Its perimeter is exactly π·w (same as a circle of diameter w, by Barbier's theorem — every curve of constant width w has perimeter π·w), while its area is smaller than a circle of the same width: A = (π − √3)·w²/2 ≈ 0.7048·w² for the triangle case, compared to π·w²/4 ≈ 0.7854·w² for a circle.

How to Use

Pick the number of sides of the base polygon (must be odd: 3, 5, 7, 9, or 11) from the dropdown.
Adjust the Width slider to scale the shape.
Enable Show construction lines to see the underlying regular polygon and the rays from each arc's centre to its two arc endpoints.
Turn on Rotate in a square to animate the Reuleaux polygon spinning while inscribed in a square — its centroid traces a small path (not a fixed point!) so that the shape stays in constant contact with all four sides of the square at every rotation angle, demonstrating constant width directly.

Did You Know?

The Reuleaux triangle is the shape at the heart of the Wankel rotary engine, where a Reuleaux-triangle rotor spins inside a specially shaped housing. Because it has constant width, a Reuleaux triangle (very slightly rounded at its corners in practice) can also be mounted in a special drilling rig to drill a hole that is almost — but not quite exactly — square, with slightly rounded corners, since the shape's centre of rotation must wobble along a small path rather than staying fixed. Several coins are based on curves of constant width built from Reuleaux-style arcs so that vending machines can recognize them by diameter regardless of orientation — the British 50 pence and 20 pence coins are 7-sided curves of constant width (though built with arcs centred on the opposite side's midpoint rather than the pure vertex-based Reuleaux construction shown here).

About this simulation

This simulator constructs genuine Reuleaux polygons — curves of constant width built from odd-sided regular polygons. For each vertex of the base n-gon, a real circular arc is drawn centred at that vertex, spanning between the two farthest ("opposite") vertices, with radius computed directly from the polygon's own geometry rather than approximated.

🔬 What it shows

Starting from a regular polygon with an odd number of sides n, n arcs are drawn, each centred at one vertex with radius equal to the width w (the vertex-to-opposite-vertex distance). Because every arc has the same radius w and each is centred on a vertex of the same polygon, the resulting curve has identical width w measured in every direction.

🎮 How to use

Choose the number of sides (3, 5, 7, 9 or 11) and adjust the Width slider to scale the shape. Toggle Show construction lines to reveal the underlying polygon and arc-centre rays, and enable Rotate in a square to watch the shape spin while its centroid traces a small path, keeping constant contact with all four sides of a square.

💡 Did you know?

The Reuleaux triangle is used as the rotor shape in Wankel rotary engines, and (with slightly rounded corners) in specialized drill bits that can cut a hole that is almost, but not exactly, square — the drill's centre must wobble along a small path rather than staying fixed in place.

Frequently asked questions

What is a curve of constant width?

It is a convex shape where the distance between any two parallel lines that just touch the shape on opposite sides (its "width" in that direction) is exactly the same no matter which direction you measure. A circle is the most familiar example, but Reuleaux polygons show that many non-circular shapes share this property too.

Why must the base polygon have an odd number of sides?

The construction relies on each vertex having a unique pair of "farthest" vertices to arc between. In an odd-sided regular polygon every vertex has exactly one such opposite pair at the maximum diagonal distance; in an even-sided polygon the farthest point from a vertex is a single vertex directly across, which does not support the same constant-width arc construction.

How is the perimeter formula π·w derived?

This follows from Barbier's theorem, which states that every curve of constant width w, regardless of its exact shape, has perimeter exactly π·w — the same as a circle of diameter w. For the Reuleaux triangle this can also be checked directly: three arcs each spanning a 60° angle of radius w sum to a half-circle's worth of arc length, equal to π·w.

Why is the area less than a circle of the same width?

Among all curves of constant width w, the circle encloses the maximum possible area (π·w²/4), while the Reuleaux triangle encloses the minimum possible area, by the Blaschke–Lebesgue theorem. For the Reuleaux triangle the exact area is (π − √3)·w²/2, noticeably smaller than the circle's area at the same width.

How does rotating in a square demonstrate constant width?

Because the shape's width is identical in every direction, it can be rotated a full turn while staying in continuous contact with all four sides of a square exactly the same size as its width. Unlike a circle, its centre of rotation is not fixed — the centroid traces a small closed path — which is why Reuleaux-based drill bits cut corners that are only almost, not perfectly, square.

⚙ Under the hood

Build genuine curves of constant width from odd-sided regular polygons (3-11 sides). Watch a Reuleaux shape rotate smoothly while inscribed in a square, drilling an almost-square hole, with live width, perimeter (pi times w) and area statistics.

Canvas 2DConstant WidthReuleaux TriangleBarbier's TheoremConvex Shapes

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install