How to Find the Circumference of a Circle

Circumference is the distance once round the outside of a circle. It is the circle’s perimeter, and the only reason it gets its own word is that you cannot add up straight sides to find it. What you can do is multiply, because the way round a circle is always the same multiple of the way across it. That multiple is π, and the entire calculation is one multiplication.

π is a measurement, not a formula

Wrap a string round a tin, mark it, straighten it against a ruler, then measure across the lid. Divide the first by the second and you get about 3.14. Do it with a bicycle wheel and you get 3.14. Do it with the equator and you get 3.14. That stubborn ratio is what π names, and it was found by measuring long before anyone could prove anything about it.

Once you see π as the answer to a measurement rather than a symbol to look up, the two formulas stop competing. C = πd says the way round is π times the way across. C = 2πr says the same, because the diameter is two radii. There is one idea here, not two.

Unrolled, every circle is three and a bit diameters

The cleanest way to understand the formula is to take the circle apart. Cut the circumference at one point and lay it out straight, then measure it against the diameter it came from. Three whole diameters fit, and then a small piece is left over.

This also explains why rough estimates work so well in the field. If you need the way round a circular tank and you have a tape across it, treble the diameter and add a bit. For a 14 unit lid that gives 42 plus a bit against a true 43.98, close enough to order material and quick enough to do while holding the tape.

A worked check you can repeat on paper

Route three is the one worth knowing, because area and circumference are often the two things a problem hands you in the wrong order. If you have the area, the circumference is twice the square root of π times the area. It agrees with the other two to every decimal shown, which is the point of running it.

How much does the approximation cost you

People reach for 22/7 or 3.14 without asking what they give up. For the circle above, here is the damage:

Value of π used	Circumference	Error
Full precision	43.9823	exact
22 over 7	44.0000	high by 0.04 percent
3.14	43.9600	low by 0.05 percent
3, for a field estimate	42.0000	low by 4.5 percent

Both of the usual approximations are wrong by about one part in two thousand, which is far below the accuracy of any tape measure. Use them without guilt on site. Do not use them in a spreadsheet that multiplies the result a thousand times, because that is where a rounding error stops being invisible.

Working backwards, and a check on the whole planet

Rearranging is straightforward. The radius is the circumference divided by 2π, and the diameter is the circumference divided by π. A circumference of 50 gives a diameter of 15.92 and a radius of 7.96.

Here is a test of that on the largest circle you have access to. The Earth measures about 40,075 km round the equator. Divide by 2π and the radius comes out at 6,378 km, which is the published equatorial radius to the kilometre. That is a satisfying way to confirm you have the rearrangement the right way up, and it is roughly the calculation Eratosthenes was doing in reverse two thousand years ago with shadows and a walking distance.

Mistakes that catch people out

• Putting the diameter into 2πr. This doubles the answer, giving 87.96 instead of 43.98. It is the single most common error, and the giveaway is that the result is almost exactly twice what it should be.
• Confusing circumference with area. C = 2πr and A = πr² look similar and answer different questions. One is a length in units, the other is a coverage in square units. Check which the question wants before you start.
• Rounding π too early. Using 3.1 rather than the calculator’s π introduces a 1.3 percent error before you have done anything else. Round the answer, never the constant.
• Answering in square units. Circumference is a distance, so it carries the plain unit. Metres, not square metres.
• Using circumference where an arc is wanted. A part of a circle is an arc, and its length is the circumference times the angle over 360. Quoting the full circumference for a quarter turn overstates it fourfold.

Try these

Common questions

What is the formula for the circumference of a circle?

C = 2πr if you have the radius, or C = πd if you have the diameter. They are the same formula, because the diameter is twice the radius. A radius of 7 gives 43.98.

Is circumference the same as perimeter?

Yes. Circumference is simply the name given to the perimeter of a circle. The word changes because you cannot get there by adding straight sides, but the quantity is the same: the distance once round the outside.

How do you find the radius from the circumference?

Divide by 2π. A circumference of 43.98 gives a radius of 7. For the diameter, divide by π instead.

Can you find the circumference from the area?

Yes, it is twice the square root of π times the area. An area of 153.94 gives a circumference of 43.98, matching the direct calculation exactly.

Is it accurate enough to use 3.14 or 22 over 7?

For practical measuring, yes. On a circle of diameter 14, 3.14 is low by 0.05 percent and 22 over 7 is high by 0.04 percent, both well inside the error of any tape. Use the calculator’s π when the result feeds further calculations.

Why is π about 3.14 and not a whole number?

Because the way round a circle genuinely is not a whole number of diameters. Three fit, and 0.1416 of another is left over. That remainder is irrational, meaning no fraction gives it exactly, which is why the digits never settle into a pattern.

Related lessons and calculators

• Area formulas for every common shape, including the circle, the sector and the ring, with a calculator that shows the working.
• Circle arc calculator, for when you need part of the way round rather than all of it, or a chord and its sagitta.
• Radius of a circle drawn round a triangle, one of the common ways a radius arrives without being measured directly.
• Radius of a circle drawn inside a triangle, the same idea from the other side.
• Area of 2D shapes calculator, for the area of the same circle without the arithmetic.
• the National Council of Teachers of Mathematics, for how circles and π are introduced in the curriculum.
• Surface area of a cone, where this circumference reappears as the curved edge of a flattened cone.
• Radius of a cylinder, where dividing a circumference by 2π is one of four ways back to a radius.
• Area of a sector, for when you only want part of the way round and the area inside it.
• Area of a semicircle, and why its perimeter is not simply half of this circumference.
• Height of a cylinder, where the circumference turns up as the width of the unrolled label.