ConverterHub

Reuleaux Triangle Area Calculator

Enter the width of the Reuleaux triangle to find its area and perimeter.

Result

70.4771
31.4159
10.00

How it is calculated

The width w of a Reuleaux triangle is the same in every direction and equals the side of the equilateral triangle it is built from. The tool computes Area = ½ × (π − √3) × w², which is about 0.7048 × w², and Perimeter = π × w, about 3.1416 × w. The area formula is the equilateral triangle (√3/4 × w²) plus three circular segments, each one sixth of a circle of radius w minus the triangle's corner piece. Values are shown to four decimals. The width must be above zero, and the area comes out in the square of the unit you entered, the perimeter in the same unit.

Worked example

Take a width of 10, in any unit such as centimetres. √3 ≈ 1.7321 and π ≈ 3.1416, so π − √3 ≈ 1.4095. Half of that is 0.7048, and multiplying by w² = 100 gives an area of 70.4771 square units. The perimeter is π × 10 = 31.4159. For comparison, a circle of diameter 10 has an area of 78.54, so the Reuleaux triangle fills about 90% of it, while its perimeter is the same as the circle's.

Try this example →

Common mistakes

  • Treating the width as the side of the shape's bounding rectangle. For this shape the width is the distance between two opposite points on the boundary, which is the same in every direction.
  • Using the equilateral triangle formula √3/4 × w². That gives 43.30 for w = 10, because it leaves out the three curved caps.
  • Entering the radius of a circle instead of the width. The arcs have radius w, so a circle of radius r with the same width has diameter 2r, and you must enter 2r.
  • Expecting the perimeter to be shorter than a circle's. It equals the circle's perimeter, π × w, though the area is smaller.
  • Mixing units. A width in centimetres gives an area in square centimetres, so convert before comparing with other shapes.

About this calculator

A Reuleaux triangle is built by taking an equilateral triangle and replacing each straight side with a circular arc centered on the opposite vertex, using the triangle’s side length as the radius. The result is a curved shape with rounded corners that has constant width — like a circle, its width is the same no matter which direction you measure it — even though it clearly isn’t round.

That constant-width property is why Reuleaux triangles show up in real engineering: the Wankel rotary engine’s rotor is shaped like one, Watts Brothers-style drill bits use the shape to bore (almost) square holes, and some coin shapes and manhole cover designs use the same principle so the shape can’t fall through its own opening at any rotation. For a given width w, its area is (π − √3) × w² / 2 and its perimeter equals π × w — the same perimeter formula as a circle of diameter w, since it’s made of three arcs each spanning 60° of a circle with that radius.

Was this helpful?

You can pick up to 3 reactions.

Comments

  • Be the first to comment.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators