★ MATH · GEOMETRY · CIRCLES
A circle with radius 5 has a circumference of 31.415927 — 2π times the radius, rounded to 6 decimal places. Start from the radius, the diameter or the area, and the calculator works out the circumference, with the formula, the step-by-step breakdown and the answer to six places. This one is the CIRCUMFERENCE tool — the distance around a circle — the perimeter-calculator page is the PERIMETER tool — the distance around rectangles, triangles and polygons — not a circle; the area-calculator page is the AREA tool — the space inside a shape — not the distance around it; and the fraction-calculator page is the FRACTION tool — adding and multiplying fractions — not lengths around a circle. No circumference calculator exists on this site — this one is the first.
The circumference is the distance around a circle: 2π times the radius, π times the diameter, or twice the square root of π times the area — three starting points, one circle, one answer. On the defaults it is a circle of radius 5, and 2π × 5 is 31.415927 to six places. π never terminates, so the display stops at six places; the breakdown shows the exact multiplication it stopped at.
It means that walking around a circle of radius 5 covers 31.415927 units — a little more than six times the radius, because the circumference is 2π times the radius and 2π is about 6.283185. The same circle measured by its diameter, 10, gives the same answer another way: π × 10 is 31.415927. And measured by its area, 78.5398163397, the answer is the same a third way: twice the square root of π times the area is 31.415927. Three starting points, one circle, one circumference — if the three ways ever disagree, one of the inputs is wrong.
π never terminates, so the answer is rounded. The true circumference of a radius-5 circle is 31.4159265359…, and the display stops at six places: 31.415927. That rounding is one unit in the sixth place, and it matters when you work backwards: dividing the displayed 31.415927 by 2π gives a radius of 5.000000 and an area of 78.539819 — not quite the 78.539816 the circle started with, because the displayed circumference was rounded up. Carry the rounded value into further working only if six places is enough.
The diameter is twice the radius — do not mix them up. The radius runs from the centre to the edge; the diameter runs all the way across, through the centre, and is twice as long. The formulas look alike — 2π × radius against π × diameter — and they give the same answer for the same circle, because 2π × 5 and π × 10 are the same multiplication. Using the diameter where the formula wants the radius doubles the answer; using the radius where it wants the diameter halves it.
The area route goes through a square root. Starting from the area means undoing the area formula first: 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. A circle of area 100 has a circumference of 35.449077 — and if the area is wrong by a little, the circumference is wrong by about half as much in proportion, because the square root halves the error.
On the defaults: a circle of radius 5.
And the other way: a circumference of 31.415927 means a radius of 5.000000, a diameter of 10.000000 and an area of 78.539819 — the reverse box finds all three from the circumference.
The defaults are a circle of radius 5, chosen so the answer is the one every textbook quotes: 2π × 5 is 31.415927. They are a starting point, not a lesson: your circle may be measured by its radius, its diameter or its area, and the calculator shows the formula it used, the step-by-step breakdown, and the verdict. The measurement must be more than 0 — a radius of 0 is not a circle — and the answer is rounded to 6 decimal places because π never terminates. Put your number in, read the answer, and check the working before you use it.
These notes repeat in every explainer section because this page is built to be read start to finish: π never terminates, so the answer is rounded; the diameter is twice the radius; and the area route goes through a square root.
| Output | Formula | What it means |
|---|---|---|
| Shape | input | Circle, radius 5. |
| Circumference | C = 2 × π × radius | 31.415927. |
| Circumference (6dp unrounded) | decimal, 6 places | 31.415927. |
| Formula | by starting point | C = 2 × π × radius. |
| Breakdown | step by step | 2 × π × 5 = 31.415927. |
| Verdict | in words | A circle with radius 5 has a circumference of 31.415927. |
| Reverse: Smallest whole radius for a circumference of 31.415926 or more | step | 5: Circle, radius 5: circumference 31.415927. |
| Reverse: Smallest whole diameter for a circumference of 62.831853 or more | step | 20: Circle, diameter 20: circumference 62.831853. |
| Reverse: Smallest whole area for a circumference of 31.415927 or more | step | 79: Circle, area 79: circumference 31.507829. |
| Reverse: Largest whole radius for a circumference of 31.415927 or less | step | 5: Circle, radius 5: circumference 31.415927. |
| Reverse: Smallest whole radius for a circumference of 100 or more | step | 16: Circle, radius 16: circumference 100.530965. |
| Reverse: Smallest whole diameter for a circumference of 100 or more | step | 32: Circle, diameter 32: circumference 100.530965. |
| Circle, radius 5 | same model | Circle, radius 5: circumference 31.415927. |
| Circle, diameter 10 | same model | Circle, diameter 10: circumference 31.415927. |
| Circle, area 78.5398163397 | same model | Circle, area 78.54: circumference 31.415927. |
| Circle, radius 1 | same model | Circle, radius 1: circumference 6.283185. |
| Circle, radius 10 | same model | Circle, radius 10: circumference 62.831853. |
| Circle, diameter 7 | same model | Circle, diameter 7: circumference 21.991149. |
| Circle, area 100 | same model | Circle, area 100: circumference 35.449077. |
| Circle, radius 2.5 | same model | Circle, radius 2.5: circumference 15.707963. |
| Circle, diameter 0.5 | same model | Circle, diameter 0.5: circumference 1.570796. |
| Circle, area 314.159265 | same model | Circle, area 314.16: circumference 62.831853. |
| Circle, radius 0 | same model | The radius must be more than 0.. |
| Circle, radius −3 (negative) | same model | The radius must be more than 0.. |
| Non-numeric input | same model | Enter a number for the radius.. |
| Circle, diameter 0 | same model | The diameter must be more than 0.. |
| Circle, area 0 | same model | The area must be more than 0.. |
| Circle, area −10 (negative) | same model | The area must be more than 0.. |
| Circle, radius 0.5 | same model | Circle, radius 0.5: circumference 3.141593. |
| Circle, diameter 1 | same model | Circle, diameter 1: circumference 3.141593. |
| Circle, area 1 | same model | Circle, area 1: circumference 3.544908. |
| Circle, radius 100 | same model | Circle, radius 100: circumference 628.318531. |
| Circle, diameter 100 | same model | Circle, diameter 100: circumference 314.159265. |
The arithmetic is exact on the numbers you type: the inputs parse exactly, and the only rounding is the six-place display of a value that never terminates. A circle of radius 5 has a circumference of 31.415927 to six places — the true value continues past the sixth decimal, so carry the 6-place value into further working only if that precision is enough. Working backwards from the displayed circumference gives a radius of 5.000000, a diameter of 10.000000 and an area of 78.539819 — the area is one unit in the sixth place off the circle’s true area, because the displayed circumference was rounded up. The calculator refuses zero or negative measurements and non-numeric input. Treat the breakdown as the check — if a measurement is wrong, the breakdown shows it.
A circle with radius 5 has a circumference of 31.415927 — 2π times the radius, rounded to 6 decimal places. A circle with diameter 10 has a circumference of 31.415927. A circle with area 100 has a circumference of 35.449077. Working backwards, a circumference of 31.415927 means a radius of 5.000000, a diameter of 10.000000 and an area of 78.539819. Smallest whole radius for a circumference of 31.415926 or more: 5. Smallest whole diameter for a circumference of 62.831853 or more: 20. Smallest whole area for a circumference of 31.415927 or more: 79. Largest whole radius for a circumference of 31.415927 or less: 5. Smallest whole radius for a circumference of 100 or more: 16. Smallest whole diameter for a circumference of 100 or more: 32. Source: Bright Side Kit, brightoolbox.com.
The circumference is 31.415927, because 2π × 5 is 31.415927. The value is rounded to 6 decimal places because π never terminates; the true value is 31.4159265359….
Multiply π by the diameter — a circle of diameter 10 has a circumference of 31.415927. The diameter is twice the radius, so π × diameter and 2π × radius are the same multiplication. A circle of diameter 7 has a circumference of 21.991149.
Take twice the square root of π times the area — a circle of area 100 has a circumference of 35.449077. 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.
The circumference is 6.283185 — 2π itself. The unit circle’s circumference is 2π, and every other circle scales from it: radius 10 is 62.831853, exactly ten times as much.
Divide the circumference by 2π — a circumference of 31.415927 means 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 true 78.539816, because the displayed circumference was rounded up.
Circumference is the distance around a circle; area is the space inside it. A circle of radius 5 has a circumference of 31.415927 and an area of 78.539816 — edging measures the circumference, paint measures the area. See the area-calculator for the inside.
Because π never terminates — its decimal goes on forever. The display stops at six places, which is enough for edging, belts and homework, and the breakdown shows the exact multiplication it stopped.
The circumference is 314.159265 — π × 100. A circle of radius 100 has a circumference of 628.318531 — exactly double, because its radius is twice the 50 of the diameter-100 circle.