Circumference Calculator
Enter any one circle measurement, radius, diameter, circumference, or area, and this calculator works out all the others instantly. Switch between metric and imperial units, or add a central angle to find arc length and sector area. Every result includes a step-by-step breakdown of the arithmetic.
Formula
Worked example
For a radius of 5 cm: C = 2 × pi × 5 = 10pi ≈ 31.4159 cm, and A = pi × 5^2 = 25pi ≈ 78.5398 cm^2. If you know the diameter is 10 cm, divide by 2 to get r = 5, same result. If you know the area is 78.54 cm^2, take sqrt(78.54 / pi) = 5. For a 90-degree arc: arc = (90/360) × 31.4159 ≈ 7.854 cm, sector area = (90/360) × 78.5398 ≈ 19.635 cm^2.
How to calculate circumference from any circle measurement
The circumference is the distance all the way around a circle, its perimeter. You can find it from any single measurement of the circle. From the radius r: C = 2 × pi × r. From the diameter d: C = pi × d (since d = 2r). From the area A: first recover the radius with r = sqrt(A / pi), then apply C = 2 × pi × r. From the circumference itself: you already have it, but you can find the radius with r = C / (2 × pi) and the area with A = pi × r^2. This calculator handles all four starting points so you never have to rearrange the formula by hand.
Arc length and sector area
A sector is a pie-slice of a circle defined by a central angle. The arc length (the curved edge of the slice) is simply the fraction of the full circumference: L = (theta / 360) × C, where theta is the angle in degrees. The sector area is the same fraction of the total area: S = (theta / 360) × pi × r^2. A 90-degree sector is a quarter circle, so its arc is C / 4 and its area is A / 4. Toggle on "Also calculate arc and sector" above to see these values for any angle from 0 to 360 degrees.
Circumference, area, and how they scale
Circumference and area answer different questions: circumference is a length (how far around), while area is the surface enclosed inside. Because area depends on the radius squared, it grows much faster than circumference as a circle gets larger. Doubling the radius doubles the circumference but quadruples the area. In practical work, fencing a circular garden depends on the circumference, while seeding or turfing it depends on the area. The constant pi (approximately 3.14159) is the fixed ratio of any circle's circumference to its diameter, the same for every circle regardless of size.
Metric and imperial units
The unit of the output is the same as the unit of the input. If you enter a radius in centimetres, the circumference and arc length come back in centimetres and the area in square centimetres. Switch the unit system selector to imperial if you are working in inches, feet, or yards, the formulas are identical. For very large or very small circles, you can enter the value in whatever unit is most convenient (millimetres, metres, kilometres, inches, miles) and convert the output separately if needed.
Circle formulas at a glance
| Quantity | Formula | Rearranged to find r |
|---|---|---|
| Diameter | d = 2r | r = d / 2 |
| Circumference | C = 2 × pi × r = pi × d | r = C / (2pi) |
| Area | A = pi × r^2 | r = sqrt(A / pi) |
| Arc length | L = (theta / 360) × C | r = L / ((theta / 360) × 2pi) |
| Sector area | S = (theta / 360) × pi × r^2 | r = sqrt(S / ((theta / 360) × pi)) |
r = radius, d = diameter, A = area, C = circumference, pi ≈ 3.14159, theta = central angle in degrees
Frequently asked questions
What is the formula for the circumference of a circle?
The circumference is C = 2 × pi × r, where r is the radius, or equivalently C = pi × d, where d is the diameter. Both give the same result because the diameter is twice the radius. Pi is approximately 3.14159.
How do I find the circumference if I only know the area?
First recover the radius: r = sqrt(A / pi). Then apply the circumference formula: C = 2 × pi × r. For example, if the area is 50 cm^2, r = sqrt(50 / pi) ≈ 3.989 cm and C = 2 × pi × 3.989 ≈ 25.066 cm. Select "Area" in the "Solve from" dropdown above to do this automatically.
How do I find the radius from the circumference?
Rearrange C = 2 × pi × r to get r = C / (2 × pi). For example, a circumference of 40 cm gives r = 40 / (2 × pi) ≈ 6.366 cm. Select "Circumference" in the "Solve from" dropdown to enter the circumference directly.
What is the difference between circumference and area?
Circumference is the distance around the edge of a circle (a length), found with 2 × pi × r. Area is the space enclosed inside the circle (a square-unit measurement), found with pi × r^2. Circumference scales linearly with the radius; area scales with the radius squared, so it grows much faster.
What is arc length and how is it different from circumference?
The circumference is the full distance around the circle. An arc is just part of that curve, defined by a central angle theta. Arc length = (theta / 360) × circumference. A 180-degree arc is a semicircle and equals exactly half the circumference. Use the arc and sector toggle above to compute this for any angle.
What is a sector and how do I find its area?
A sector is a pie-slice of a circle bounded by two radii and an arc. Its area is S = (theta / 360) × pi × r^2, where theta is the central angle in degrees. A 90-degree sector (a quarter circle) has an area of pi × r^2 / 4. This calculator outputs sector area when you enable the arc and sector toggle.