Circumference Of Circle Calculator

What this calculator does

You have something round, and you need the distance all the way around the outside of it. The strip of edging for a circular flower bed. The band that wraps a pipe or a tank. How far a wheel travels in one full turn. That distance around the rim is the circumference.

Measure the radius, the distance from the centre straight out to the edge, type it in, and pick the unit it is in. The tool hands back the circumference in that same unit.

Using the calculator

1. Type the radius into the box.
2. Pick the unit it is measured in (millimetres, centimetres, metres, kilometres, inches, feet, yards or miles).
3. Press Calculate.

The circumference comes back in the same unit you chose, since it is a distance and not an area. The radius has to be a positive number, because a real circle cannot have zero or negative width.

The formula | circumference = 2 × π × radius

The circumference of a circle is:

circumference = 2 × π × radius

Double the radius to get the full width across the middle (the diameter), then multiply by π.

So what is π doing here? This is the one place where π is not just a number you borrow, it is the answer. π is defined as the distance around a circle divided by the distance straight across it. Every circle, whatever its size, is almost exactly 3.14159 diameters around. That ratio is π. So the circumference is π diameters, and since a diameter is two radii, that comes to 2 × π × radius.

π itself never ends and never repeats, which makes it an irrational number, something Johann Lambert proved back in 1768. The tool uses your browser's built-in value of π, good to about 15 to 16 digits, and works the whole thing out on your device.

Why the answer is a length, not an area

Here is the thing people mix up. The circumference is a length, so it comes out in the same unit as the radius. A radius in centimetres gives a circumference in centimetres, not square centimetres. That squared unit belongs to area, where you multiply a length by a length. The circumference is just one trip around the edge, so the unit stays plain.

The answer is rounded to two decimal places, which is plenty for marking out a length or cutting a band to size.

A worked example with a 5 cm radius

Say the radius is 5 cm.

1. Double it to get the diameter: 2 × 5 = 10 cm.
2. Multiply by π: 10 × π = 31.4159...

So the circumference is about 31.42 cm, which is what the tool shows.

A nice way to picture it: the rim is a little over three times as long as the width straight across. Three diameters and a bit, every single time.

If you have the diameter instead

Measured straight across the middle rather than out from the centre? That is the diameter, and you have two easy options. Halve it to get the radius (radius = diameter ÷ 2) and type that in, or skip the step and use the diameter form of the formula directly:

circumference = π × diameter

Both give the same answer, since the diameter is simply twice the radius. If you would like the area inside the circle as well, the area of circle calculator handles that.

Circumference for common radii

A few common radii worked out. The circumference is in whatever unit you measured the radius in.

Questions people ask

What is the circumference of a circle with a radius of 5?

About 31.42, in whatever unit you measured the radius. Double the radius and multiply by π: 2 × π × 5.

Is the formula 2πr or πd?

Both, and they give the same answer. Use 2 × π × radius when you have the radius, and π × diameter when you have the diameter, since the diameter is twice the radius.

How do I find the circumference from the diameter?

Multiply the diameter by π. So circumference = π × diameter.

Is the circumference in square units?

No. It is a length, so it stays in the same unit as the radius. Square units are for area, where a length is multiplied by a length.

What is the difference between circumference and perimeter?

They are the same idea, the distance around the edge of a shape. "Perimeter" is the usual word for straight-sided shapes, while "circumference" is the word kept for circles.

References

A quick note on where this comes from. The whole idea rests on one fact: the distance around any circle, divided by the distance across it, is always the same number, π. That ratio is the definition of π, tabulated by the US National Institute of Standards and Technology. That π never ends or repeats was proved by Johann Lambert in 1768. And to show how far the formula reaches, Eratosthenes used it to measure the size of the whole Earth in around 240 BCE, from little more than shadows and geometry. For further reading, see Circumference.

1. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value and definition of π as a circle's circumference-to-diameter ratio. https://dlmf.nist.gov/
2. J. H. Lambert (1768), the first proof that π is an irrational number.
3. Eratosthenes (c. 240 BCE), the first known measurement of the Earth's circumference using the geometry of circles.