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.
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.