Geometry
How To Calculate The Circumference Of A Circle
Circumference is the distance around a circle, and it is the one circle measurement you can take with a piece of string, which makes it the easiest to check.
Quick Answer
circumference = 2 x pi x radius
- C
- Circumference, the distance around the circle
- r
- Radius, from centre to edge
- d
- Diameter, twice the radius
Multiply the radius by two to get the diameter, then multiply by pi. A circle of radius seven has a circumference of about 43.98 units. If you know the diameter instead, skip the doubling and multiply it by pi directly.
What Is The Circumference Of A Circle?
The circumference of a circle is the distance all the way around its edge. It is found by multiplying the diameter by pi, or equivalently by multiplying twice the radius by pi. A circle of radius seven has a circumference of about 43.98.
Pi is the ratio of a circle's circumference to its diameter, and it is the same number for every circle ever drawn. That is the remarkable part: a coin and a planet both have a circumference of about 3.14159 times their diameter, with no dependence on size.
Because pi is irrational, the circumference is almost never a tidy number. Working to four decimal places, as most engineering and school work does, is far more precision than any physical measurement justifies.
The diameter form is the easier one to remember. If you can measure straight across the widest part of the circle, the circumference is simply that measurement times pi. The radius form is the same statement with the doubling moved inside.
Circumference and area are different quantities and are constantly confused. Circumference is a length, measured in metres or inches. Area is a surface, measured in square units. A circle of radius seven has a circumference of 43.98 and an area of 153.94, and the two numbers are not comparable.
Rearranging the formula solves the reverse problem. Dividing the circumference by pi gives the diameter, and halving that gives the radius. This is exactly how you find the size of a pipe or a tree trunk when you can only wrap a tape measure around it.
The most practical use is rolling distance. A wheel travels its own circumference in one full rotation, so the number of rotations needed to cover a distance is the distance divided by the circumference. A wheel of radius seven centimetres has a circumference of about 43.98 centimetres, so it turns about 22.74 times in a kilometre.
That relationship is how odometers and speedometers have worked for a century. The instrument counts rotations and multiplies by the known circumference of the tyre, which is why fitting a different tyre size makes the speedometer read incorrectly.
Units work straightforwardly. A radius in centimetres gives a circumference in centimetres, with no squaring and no cubing. If the answer is needed in metres, divide by a hundred at the end rather than converting the radius first.
A semicircle's arc is half the circumference, and the perimeter of a semicircular shape adds the straight diameter back on. That extra straight edge is easy to forget, and it is the reason a half circle's boundary is not simply half of the full circle's.
Finally, remember that pi is not 3.14 in any serious context. The extra digits matter when the radius is large: a one percent error in pi becomes a one percent error in the circumference, which on a hundred kilometre measurement is a whole kilometre.
Formula
C = 2 x pi x r
The standard form. Double the radius and multiply by pi.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| C | Circumference | metres | Distance around the circle. |
| r | Radius | metres | Centre to edge distance. |
C = pi x d
The same formula with the doubling already done, since the diameter is twice the radius.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| C | Circumference | metres | Distance around the circle. |
| d | Diameter | metres | Distance straight across the widest part. |
rotations = distance / circumference
A wheel advances one circumference per turn, which is how odometers and speedometers work.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| n | Rotations | turns | Number of full turns of the wheel. |
| D | Distance | metres | Distance to be covered. |
| C | Circumference | metres | Rolling circumference of the wheel. |
How To Calculate The Circumference Of A Circle
- 1
Find the radius or the diameter
They are related by a factor of two, so either works. If you measured across the circle, you have the diameter and can skip straight to multiplying by pi.
- 2
Check both measurements use the same unit
Pi has no units, so the answer inherits whatever unit the radius was in. Mixing centimetres and metres is the usual source of an answer out by a factor of a hundred.
- 3
Double the radius
A radius of seven gives a diameter of fourteen. This step exists only to convert to diameter, which is the quantity pi actually relates to.
- 4
Multiply by pi
Fourteen times 3.14159 is about 43.9823. Using 3.14 instead gives 43.96, which is fine for a rough estimate but drifts as the circle gets bigger.
- 5
Decide whether the answer is plausible
The circumference must be a little more than three times the diameter, always. If your answer is less than the diameter, or more than four times it, something has gone wrong.
Examples
Example 1: A circle of radius seven
- Radius
- 7
| Step | Calculation | Result |
|---|---|---|
| Diameter | 2 x 7 | 14 |
| Area | pi x 7^2 | 153.938 |
| Circumference | 2 x pi x 7 | 43.9823 |
Result: A circle of radius seven has a circumference of 43.9823 units and encloses an area of 153.938 square units.
Example 2: A wheel of radius 0.35 metres
- Radius
- 0.35 m
| Step | Calculation | Result |
|---|---|---|
| Diameter | 2 x 0.35 | 0.7 |
| Area | pi x 0.35^2 | 0.3848 |
| Circumference | 2 x pi x 0.35 | 2.1991 |
Result: A wheel of radius 0.35 metres rolls 2.1991 metres per turn, so it rotates about 454.7 times over a kilometre.
Example 3: A circular flower bed of radius ten metres
- Radius
- 10 m
| Step | Calculation | Result |
|---|---|---|
| Diameter | 2 x 10 | 20 |
| Area | pi x 10^2 | 314.1593 |
| Circumference | 2 x pi x 10 | 62.8319 |
Result: A flower bed of radius ten metres needs 62.8319 metres of edging, and covers 314.1593 square metres of ground.
Calculator
Circumference
43.9823
- Area
- 153.938
- Diameter
- 14
- Rotations per kilometre
- 22.7364
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Circumference Of A Circle calculator page.
Common Mistakes
Multiplying the radius by pi without doubling
That gives half the circumference, which is the length of a semicircular arc rather than the whole circle. The factor of two is the most commonly dropped part of the formula.
Using the diameter where the radius belongs
The two forms are not interchangeable. Doubling an already doubled value gives twice the correct answer, and the result still looks like a plausible length.
Reporting the answer in square units
Circumference is a length, so the unit is plain metres or inches. Square units describe the area, and mixing them up is the classic circle error.
Rounding pi too early
Using 3.14 is fine for a quick estimate, but rounding mid-calculation and then multiplying again compounds the error. Keep the full value until the final line.
Forgetting the diameter on a semicircle
The boundary of a half circle is half the circumference plus the straight diameter. Leaving out the straight part understates the perimeter badly.
Assuming a circumference tells you the area directly
The two are related through the radius, not directly. Divide by two pi to recover the radius first, then use it in the area formula.
Measuring the diameter off centre
A chord measured anywhere other than through the centre is shorter than the diameter. Measure at the widest point, or wrap a tape around the circle instead.
FAQ
What is the formula for the circumference of a circle?
Two times pi times the radius, or pi times the diameter. Both describe the same distance; the second just has the doubling already included.
How is circumference different from area?
Circumference is the distance around the edge, measured in plain units. Area is the surface inside, measured in square units. A circle of radius seven has a circumference of 43.98 and an area of 153.94.
How do you find the radius from the circumference?
Divide the circumference by two pi. A circumference of 43.98 gives a radius of seven, which you can check by multiplying back.
Why is pi in the formula?
Because pi is defined as the ratio of a circle's circumference to its diameter. It is the same for every circle, which is what makes the formula universal.
How many times does a wheel turn in a kilometre?
Divide one thousand metres by the wheel's circumference. A wheel of radius 0.35 metres rolls 2.1991 metres per turn, so about 454.7 turns per kilometre.
Does the formula work for a semicircle?
Half the circumference gives the curved arc. If you need the full boundary, add the straight diameter as well, because a semicircle has a flat edge.
References
- [1]Wikipedia, Circumference — https://en.wikipedia.org/wiki/Circumference
- [2]Wikipedia, Pi — https://en.wikipedia.org/wiki/Pi
- [3]Wikipedia, Circle — https://en.wikipedia.org/wiki/Circle
- [4]Math is Fun, Circumference of a circle — https://www.mathsisfun.com/geometry/circle.html