Reuleaux triangle

This shape looks like a triangle.

But its sides are curved. It has a special trick. It stays the same width all around.

This means it can turn inside a square. It will touch all four sides. It cannot fall through a round hole. This is why it works for manhole covers.

Many people used this shape long ago. A man named Leonardo da Vinci used it. You can see it on a guitar pick.

It is a very cool shape.

A Reuleaux triangle looks like a curved triangle.

It is a special shape with constant width. This means its width is always the same. No matter how you turn it, it stays the same size from side to side.

Because of this, it can spin inside a square. It will touch all four sides as it turns. This shape is named after Franz Reuleaux. He was a German engineer from the 1800s. Other smart people knew it too. Leonardo da Vinci used it for a map. You can also find this shape on guitar picks.

It is even used for drill bits. These bits can make square holes. The Reuleaux triangle has very sharp corners. Each corner has an angle of 120 degrees. This is the sharpest angle for any shape with constant width. It also has the smallest area of all such shapes. Even with its curves, it is very useful in many machines.

Imagine a shape that looks like a triangle, but its sides are smooth curves instead of straight lines.

This is called a Reuleaux triangle. It has a very special property called constant width. This means if you place the shape between two flat, parallel lines, the distance between those lines stays exactly the same no matter how you turn it.

Most people think only a circle has this trick. However, the Reuleaux triangle proves that other shapes can work too. It is the simplest and most famous shape of this kind. Because it stays the same width, it can do amazing things in machines.

You can make this shape using just a compass. First, you mark two points on a piece of paper. You draw a circle using one point as the center that passes through the second point. Then, you draw another circle of the same size using the second point as the center. Finally, you draw a third circle using a new center point where the first two circles meet. The curved shape left in the middle is your Reuleaux triangle. You can also build it by starting with a perfect equilateral triangle and drawing arcs between the corners. Each arc is centered on one corner and connects the other two.

Many clever people studied this shape long before it was named. Leonardo da Vinci used a version of this shape for a map projection. The mathematician Leonhard Euler also studied these shapes in the 1700s. He called them "orbiforms." Later, a German engineer named Franz Reuleaux studied how machines move. He used these shapes in his designs for translating motion. This is why we call the shape by his name today. He was a pioneer in studying how different parts of a machine work together.

The Reuleaux triangle is a shape of extremes.

It has the smallest area of any shape that has a constant width. It also has the sharpest corners, which are exactly 120 degrees. This is the smallest angle possible for any constant-width shape. Even though it is pointy, it can still spin perfectly inside a square. As it rotates, it touches all four sides of the square at once. It covers about 98.8% of the square's area, leaving only tiny gaps near the corners. It is a very efficient shape for moving inside tight spaces.

You can find this math in action all around you.

Many guitar picks are shaped like a Reuleaux triangle. It is also used for the nuts on fire hydrants. Some special drill bits use this shape to carve out square holes. Even some company logos and road signs use these curved triangles in their designs. It shows how a simple math idea can become a tool for engineers and artists alike. From music to city streets, the Reuleaux triangle is always working.

A Reuleaux triangle is a unique geometric shape that resembles a triangle with curved sides.

While it looks like a triangle, it possesses a mathematical property called constant width. This means the distance between any two parallel lines touching its edges remains identical, regardless of the shape's orientation. This property makes it a member of a special group known as curves of constant width. Most people assume only circles have this capability, but the Reuleaux triangle is the simplest and most famous non-circular example. This characteristic allows it to function in specialized mechanical roles where a circle might not be ideal.

To understand its mechanism, we can look at how it is constructed. One method uses a compass to draw three intersecting circles of equal radii. First, you mark two points and draw a circle centered at one that passes through the other. You then draw a second circle of the same size centered at the second point. Finally, a third circle is drawn using one of the intersection points as its center. The central area where all three circles overlap forms the Reuleaux triangle. Alternatively, you can start with a perfect equilateral triangle and draw circular arcs between its vertices. Each arc must be centered on one vertex and connect the remaining two.

There are several ways to view the mathematical properties of this shape. The most fundamental is the constant width, which is defined by parallel supporting lines.

When these lines touch the shape, one line will always contact a vertex. The opposite line will touch the arc directly across from that vertex. The distance between these lines always equals the radius of the arc used in construction. This constant distance is why the shape can rotate within a square while always touching all four sides. This specific ability has earned the shape the nickname of a "Reuleaux rotor."

The history of this shape involves many famous thinkers.

While named after the 19th-century German engineer Franz Reuleaux, the shape was known much earlier. Leonardo da Vinci utilized the shape for a map projection. The mathematician Leonhard Euler studied these shapes in the 18th century, specifically in his 1781 publication "De curvis triangularibus." Euler referred to these constant-width shapes as "orbiforms." Reuleaux himself was a pioneer in studying how machines translate different types of motion. He used these triangles in his engineering designs to solve specific mechanical problems.

In mathematics, the Reuleaux triangle is considered an extremal shape, meaning it sits at the edge of what is possible.

According to the Blaschke–Lebesgue theorem, it has the smallest possible area of any curve with a given constant width. It also features the sharpest possible corners for a constant-width shape, with angles of exactly 120 degrees. Despite its pointiness, it can rotate inside a square and cover approximately 98.8% of that square's area. However, it cannot reach the very corners of the square, leaving small uncovered gaps. Furthermore, it is noted for being highly asymmetrical compared to a circle.

Practical applications of the Reuleaux triangle are surprisingly common in daily life.

You can find this shape in the design of many guitar picks. It is also used for the nuts on fire hydrants and in certain types of pencils. Engineers use special Reuleaux-shaped drill bits to create filleted square holes. Because of its unique geometry, it can even be used as a manhole cover to prevent it from falling through its own opening. Beyond tools, its distinct look is often used in graphic design for corporate logos and various signs.

The concept can even be expanded into higher dimensions. One way to generalize it is through the Reuleaux tetrahedron, which is formed by the intersection of four balls.

Interestingly, a standard Reuleaux tetrahedron does not actually have constant width. To fix this, mathematicians create a Meissner tetrahedron by rounding its edges. Another method for three-dimensional expansion is to create a surface of revolution from the 2D triangle. These advanced shapes continue to help mathematicians understand the relationship between symmetry, area, and motion in complex systems.