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Circumference Calculator

Any one measurement of a circle gives the rest, with pi kept as pi.

By Nethanel Bar, Co-founder & CEO

Last updated

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One number decides a circle completely

A circle has only one degree of freedom, so any single measurement fixes all the others. From the radius r: the diameter is 2r, the circumference is 2πr, and the area is πr². Those three formulas are the whole subject, and the only thing to keep straight is that the circumference uses the radius once while the area uses it twice - squared, not doubled.

π is what makes a circle a circle: it is the ratio of any circumference to its own diameter, the same number for every circle that has ever been drawn. It is irrational, so it has no exact decimal - which is precisely why this page keeps it as the letter π. A circumference of 6π is an exact answer; 18.85 is a rounding, and the two are not interchangeable when a mark scheme asks for one of them.

That exactness works backwards too. Give a circumference of 10 and the radius is 5/π - still exact, still a real number - and the area comes out as 25/π. Every competing calculator multiplies by 3.14159 on the first step and can therefore never hand back the answer a school question wants.

What to notice

  • C = 2πr and C = πd are the same formula, because d = 2r. Use whichever measurement you were given rather than converting first.
  • The area squares the radius; the circumference does not. Doubling a circle's radius doubles its circumference but quadruples its area.
  • π is a ratio, not a length: it is circumference divided by diameter, and it comes out the same for every circle. That is the fact the whole subject rests on.
  • An answer of 6π is exact and 18.85 is not. Keep the π unless the question asks for a decimal or gives you a value of π to use.
  • Going backwards from a circumference divides by 2π, which leaves π in the denominator. 5/π is an ordinary number, roughly 1.59.
  • The units follow the measurement: a radius in centimetres gives a circumference in centimetres and an area in square centimetres.

How to find a circumference

  1. Identify what you have been given

    Radius, diameter or circumference. They are easy to mix up in a worded question, and using a diameter where the formula wants a radius doubles the answer.

  2. If it is a diameter, halve it

    r = d/2. Doing this first means you only ever need the one formula rather than remembering two.

  3. Multiply by 2π

    C = 2πr. Leave π as a letter for now - 2π · 3 is 6π, and that is a finished exact answer.

  4. Only then reach for a decimal

    If the question wants a number, multiply by 3.14159… at the very end. Rounding early loses precision that the rest of a multi-step problem needs.

  5. To go backwards, divide

    From a circumference, r = C/(2π). The π stays in the denominator, which is exact; dividing by 3.14 here is where accuracy is usually thrown away.

Circles by the numbers

The same four measurements for common radii. The exact column is the answer; the decimal is a rounding of it.

RadiusDiameterCircumferenceAreaC as a decimal
122ππ6.2832
244π4π12.5664
366π9π18.8496
51010π25π31.4159
71414π49π43.9823
102020π100π62.8319
1/21ππ/43.1416
2.555π6.25π15.708

Worked examples

From a radius of 3

plain
r = 3

C = 2π · 3 = 6π exactly, which is about 18.8496. The area is π · 3² = 9π, about 28.27. Notice the radius is squared for the area and not for the circumference - that single difference is most of what goes wrong on this topic.

From a diameter of 10

plain
d = 10

Halve it first: r = 5. Then C = 2π · 5 = 10π, and A = π · 5² = 25π. Using the diameter directly in C = 2πr would have given 20π - exactly twice too big, and the most common error here.

Backwards from a circumference of 10

plain
C = 10

r = 10/(2π) = 5/π, which is exact and roughly 1.5915. The area is then π · (5/π)² = 25/π, about 7.96. Keeping π in the denominator is what makes these exact; dividing 10 by 6.28 first would have rounded before the area was even started.

A fractional radius: 1/2

plain
r = 1/2

C = 2π · 1/2 = π exactly - a circle of radius one half has a circumference of exactly π. Its area is π/4. Fractions stay fractions, so nothing is lost on the way.

Why 6π is not 18.85

plain
r = 3

6π is a single exact number. 18.8496 is a decimal that agrees with it to four places and then differs forever, because π is irrational. If a question says "give your answer in terms of π", 6π is the answer and 18.85 is marked wrong.

Common mistakes

  • Putting a diameter into C = 2πr. That doubles the answer; halve it to a radius first, or use C = πd instead.
  • Squaring the radius for the circumference. C = 2πr uses r once; only the area squares it.
  • Rounding π at the start. Multiply out at the very end if a decimal is wanted, so intermediate steps stay exact.
  • Using 22/7 as if it were π. It is a rough approximation, out by about a thousandth - fine for mental arithmetic, wrong for an exact answer.
  • Forgetting the units square for area. A radius in metres gives a circumference in metres and an area in SQUARE metres.
  • Confusing circumference with perimeter of some other shape. Circumference is specifically the way round a circle - a straight-sided figure has a perimeter instead.

Circumference FAQ

What is the formula for circumference?
C = 2πr from the radius, or the identical C = πd from the diameter, since a diameter is two radii. For a radius of 3 that is 6π, which is about 18.85. Use whichever measurement you were actually given.
How do I find the radius from the circumference?
Divide by 2π: r = C/(2π). A circumference of 10 gives r = 5/π, roughly 1.59. Keeping π in the denominator makes the answer exact - dividing by 3.14 first rounds before you have finished.
What is the difference between circumference and area?
The circumference is the distance round the edge and uses the radius once: 2πr. The area is the space inside and uses it twice: πr². That is why doubling a radius doubles the circumference but quadruples the area.
What does "in terms of pi" mean?
Leave π as a symbol instead of multiplying it out: 6π rather than 18.85. It is the exact answer, and most exam questions ask for it because it can be checked without a calculator - and because 18.85 is only an approximation.
Is 22/7 the same as pi?
No. 22/7 is 3.142857…, while π is 3.141592…, so they differ from the third decimal place onwards. It is a handy mental approximation and a wrong answer whenever exactness matters; π is irrational and has no exact fractional value at all.
Why is pi the same for every circle?
Because all circles are the same shape scaled up or down, and scaling multiplies the circumference and the diameter by the same factor - so their ratio never changes. That constant ratio IS π, which is why it turns up in every circle formula.
How do I find the diameter from the circumference?
Divide by π: d = C/π. It is one step rather than two, since C = πd directly. A circumference of 10 gives a diameter of 10/π, about 3.18 - and half of that is the radius.
Does this calculator take the area as an input?
No, deliberately. From an area, r = √(A/π), which is neither a fraction nor a fraction times a power of π - so accepting it would mean either a rounded answer or a fake exact one. The area is always reported; it is just not an input.

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