A shape often referred to as a "triangle with rounded corners" is formally known as a Reuleaux triangle.
Based on the provided reference, a Reuleaux triangle is defined as a curved triangle with constant width. It stands out as the simplest and most recognized curve with constant width other than the circle itself. This distinctive shape is created geometrically from the intersection of three circular disks. Each of these disks has its center positioned precisely on the boundary of the other two disks. This construction inherently results in the characteristic rounded vertices or "corners."
While it has three vertices like a standard triangle, these vertices are not sharp points but are smooth, curved arcs.
Key Properties of a Reuleaux Triangle
Understanding the unique attributes of this shape helps clarify why it's not a typical triangle:
- Curved Sides: Unlike traditional triangles formed by straight lines, a Reuleaux triangle is bounded by three circular arcs.
- Constant Width: This is its most remarkable property. No matter how you measure the distance between two parallel tangent lines touching the shape, the width is always the same. This is analogous to a circle.
- Formation: As mentioned, it's constructed from intersecting circles, giving it its rounded form.
- Vertices: It has three points often referred to as vertices, but they are where the curved arcs meet, resulting in a rounded rather than pointed corner.
Why "Curved Triangle"?
The term "curved triangle" is used because it retains some characteristics of a triangle (three bounding arcs/sides, three vertices) but is defined by curves, not straight lines. The rounded corners are a direct consequence of its construction from intersecting circles.
Comparing Shapes: Triangle vs. Reuleaux Triangle
| Feature | Standard Triangle | Reuleaux Triangle |
|---|---|---|
| Sides | 3 straight segments | 3 circular arcs |
| Corners | 3 sharp points | 3 rounded vertices |
| Width | Varies depending on orientation | Constant |
| Perimeter | Sum of side lengths | Longer than circle/triangle of same width |
| Area | Formula based on base/height | Smaller than circle of same width |
Applications of the Reuleaux Triangle
Despite its unusual shape, the Reuleaux triangle has practical uses:
- Coin Design: Some coins, like the British 20p and 50p pieces, are based on the Reuleaux heptagon (a shape with constant width based on seven curves), which is a generalization of the Reuleaux triangle. Their constant width allows them to be used in vending machines.
- Drill Bits: Reuleaux triangle-shaped drill bits can drill holes that are nearly square (with slightly rounded corners).
- Mechanical Engineering: Used in cams and other mechanisms where constant width motion is required.
In summary, when people refer to a "triangle with rounded corners," especially in a geometric context, they are likely describing a Reuleaux triangle, characterized by its curved sides, rounded vertices, and fascinating constant width property.
