A Reuleaux triangle is a rounded shape built from an equilateral triangle by replacing each side with a circular arc centered at the opposite vertex. Its most surprising property is constant width, which means the distance between two parallel supporting lines stays the same no matter how the shape is turned. This makes it behave in some ways like a circle even though it has corners and curved sides.
Studying it connects geometry, measurement, motion, and design.
Understanding Geometry: Reuleaux Triangles
Constant width is best understood by thinking about a pair of straight walls that can move closer together. Place the shape between them, with each wall just touching it. As the shape turns, the touching points move from an arc to a vertex, yet the wall spacing does not change.
The reason comes from the matching arcs. When one wall touches an arc, the opposite wall reaches the vertex that was used as the centre for that arc.
The distance from that vertex to every point on the arc is the same radius. This relationship holds in each direction because the three parts of the shape are arranged evenly.
This is different from merely having a round outline. An oval may look smooth, but its width changes when it turns. A rectangle has one width when it rests on a long side and a different width when it rests on a short side.
A shape of constant width must pass a tougher test. Every pair of parallel tangent lines must remain equally far apart. A circle passes because every diameter has the same length.
The Reuleaux triangle passes for a less obvious reason, using arcs and opposite vertices rather than diameters. This makes it a useful example of how a geometric property can depend on all directions, not just a few measurements.
Rolling reveals an important limit of constant width. Between two parallel rails, the shape can keep touching both rails throughout a turn. On one flat surface, however, its motion is not as smooth as the motion of a wheel.
Its centre rises and falls as different arcs meet the ground. A circle has one fixed radius from its centre to the ground, so its centre travels horizontally. For the Reuleaux triangle, that distance changes during the turn.
This is why constant width does not mean constant radius. Students often mix up these ideas.
Width concerns opposite supporting lines. Radius concerns distances measured from one chosen centre.
Engineers have used constant width shapes where a part must fit a fixed gap while rotating. Some drill bits use a Reuleaux triangle based form to make holes that are nearly square, though the corners need special guidance for a true square. Shapes related to this one can appear in coins, mechanical gauges, and design puzzles.
When studying the geometry, sketch the original equilateral triangle first. Mark the centre of each arc clearly. Then trace one arc while keeping a compass opening equal to the triangle side.
For measurements, separate arc length from straight line length. For motion, draw the supporting lines and identify the contact points. These diagrams make the constant width argument much easier to follow.
Key Facts
- A Reuleaux triangle is formed from an equilateral triangle of side length s using three circular arcs of radius s.
- The width of a Reuleaux triangle is constant and equals the side length of the original equilateral triangle: w = s.
- Its perimeter is the length of three 60 degree arcs: P = pi s.
- Its area is A = ((pi - sqrt(3))/2)s^2.
- A Reuleaux triangle can roll between two parallel lines while keeping constant contact with both lines.
- Because it is not a circle, its center does not stay at a constant height while rolling on a flat surface.
Vocabulary
- Reuleaux triangle
- A Reuleaux triangle is a curve of constant width made from three circular arcs based on an equilateral triangle.
- Curve of constant width
- A curve of constant width has the same distance between parallel tangent lines in every direction.
- Equilateral triangle
- An equilateral triangle is a triangle with three equal sides and three equal 60 degree angles.
- Circular arc
- A circular arc is a connected part of the circumference of a circle.
- Supporting line
- A supporting line touches a shape at its boundary while the whole shape lies on one side of the line.
Common Mistakes to Avoid
- Thinking a Reuleaux triangle is just a triangle with rounded corners. This is wrong because each curved side is a precise circular arc centered at the opposite vertex.
- Assuming constant width means constant radius from a center point. This is wrong because the Reuleaux triangle does not have one fixed center like a circle.
- Using the area formula for an equilateral triangle alone. This is wrong because the Reuleaux triangle includes curved circular segments outside the original triangle.
- Believing it rolls exactly like a wheel with a fixed axle height. This is wrong because its width stays constant, but its center moves up and down as it rolls.
Practice Questions
- 1 A Reuleaux triangle is constructed from an equilateral triangle with side length 8 cm. What is its constant width and perimeter?
- 2 Find the area of a Reuleaux triangle made from an equilateral triangle with side length 6 cm. Use A = ((pi - sqrt(3))/2)s^2 and give an approximate answer.
- 3 Explain why a Reuleaux triangle can fit snugly between two parallel lines while rotating, but still does not make an ideal circular wheel for a bicycle.