How to calculate circumference, step by step
To calculate a circumference, multiply 2π by the radius. That is the whole method, and the two other routes below are just the same multiplication with the radius found a different way. A circle of radius 5 has a circumference of 31.415927, because 2 × π × 5 is 31.415927. A circle of radius 2 has a circumference of 12.566371, because 2 × π × 2 is 12.566371. And a circle of radius 3 has a circumference of 18.849556, because 2 × π × 3 is 18.849556.
The worked example: a circle of radius 5
The worked example. Step 1 — take the radius: 5. Step 2 — multiply by 2: 2 × 5 is 10. Step 3 — multiply by π: 10 × π is 31.415927 to six places. Step 4 — write the answer with its units: the circumference is 31.415927, in whatever units the radius was measured in. Step 5 — check it by the diameter: the diameter is 10, and π × 10 is 31.415927 — the same answer, two ways. The six places are where the display stops, not where π stops: the true value is 31.4159265359…, and it goes on forever.
From the diameter: multiply π by the diameter
If the measurement in hand is the diameter — the distance all the way across, through the centre — the formula is C = π × diameter. Take a circle of diameter 14. Step 1 — take the diameter: 14. Step 2 — multiply by π: 14 × π is 43.982297 to six places. The circumference is 43.982297. The diameter is twice the radius, so this is the same multiplication as before with the 2 moved: π × 14 is the same as 2π × 7, and a circle of diameter 14 is a circle of radius 7. The circumference-calculator takes the diameter as one of its three starting points.
From the area: twice the square root of π times the area
If the measurement in hand is the area, the route goes through a square root: the area is π times the radius squared, so the radius is the square root of the area divided by π, and the circumference is twice π times that — C = 2 × √(π × area). Take a circle of area 50. Step 1 — multiply the area by π: 50 × π is 157.079633. Step 2 — take the square root: √157.079633 is 12.533141. Step 3 — double it: 2 × 12.533141 is 25.066283. The circumference is 25.066283. A circle of area 25 has a circumference of 17.724539, by the same three steps.
Working backwards: from a circumference to the radius
Sometimes the circumference is the measurement in hand and the radius is the question. Divide the circumference by 2π: a circumference of 31.415927 divided by 2π is a radius of 5.000000, a diameter of 10.000000 and an area of 78.539819. The area comes out one unit in the sixth place off the circle’s true area of 78.539816, because the circumference it started from was rounded up — the display stopped at six places, and the rounding rides along. For a round target, a circumference of 50 means a radius of 7.957747 — divide 50 by 2π — and a diameter of 15.915494.
Checking by measuring
Whatever method you used, check it by measuring: wrap a tape around the circle, or measure the diameter and multiply by π. For a circle of radius 5, the tape should read about 31.415927, and the diameter should read 10 — and π × 10 is 31.415927, the same as 2π × 5. If the tape and the multiplication disagree, one of the measurements is wrong: the most common mistake is measuring the diameter and using it as the radius, which doubles the answer, or measuring the radius and using it as the diameter, which halves it. Run your numbers through the circumference-calculator and compare them with the worked example above.
Common mistakes
The first is mixing up the radius and the diameter: the radius runs from the centre to the edge, the diameter runs all the way across, and the diameter is twice the radius — using one where the formula wants the other doubles or halves the answer. The second is using the area where the formula wants the circumference, or the other way round: a circle of radius 5 has a circumference of 31.415927 and an area of 78.539816, and the two numbers answer different questions — edging measures the circumference, paint measures the area. The third is rounding π too early: using 3.14 for π gives 31.4 for a radius-5 circle, which is off by 0.015927 — fine for an estimate, wrong for a cut. And the fourth is forgetting that the answer is rounded: the display stops at six places, and working backwards from the displayed value gives an area of 78.539819, not quite the true 78.539816.
A quick check you can do in your head
You can sanity-check any circumference without a calculator. First, remember that the circumference scales with the radius: a radius of 2 means 12.566371 and a radius of 3 means 18.849556 — double the radius and the circumference doubles too, so a radius of 5 means 31.415927. Second, use the diameter as a cross-check: π × diameter must agree with 2π × radius, and if the two ways disagree, one of the numbers is wrong. If a calculator says the circumference of a radius-5 circle is far from 31.415927, the numbers are wrong.