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CalcMax

Circumference Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

31.4159 cm

Circumference

A circumference calculator takes whichever of the two widths you have to hand — the radius, measured from the centre out to the edge, or the diameter, measured straight across — and returns how far it is around the circle once, which is to say its perimeter, in centimetres. The formula is 2πr, twice the radius multiplied by pi, and it is the oldest of the circle formulas: this is the one the Greeks were computing by exhausting polygons, and the one that gave pi its name in the first place, since pi is defined as the circumference of any circle divided by its diameter. That definition is worth holding on to, because it is what makes the reverse direction easy. Multiply a diameter by pi and you have the distance around; divide a circumference by pi and you have the diameter back. The page's own arithmetic goes the long way round on purpose, halving the diameter to a radius and then doubling it again inside the formula, and the reason is that the two spellings of the same expression do not always land on the same last digit when a computer does them. The page takes the route that agrees with its sibling pages, so a circle of diameter 12 reads 37.6991 here and 37.6991 on the overview page. Only one of the two boxes is to be filled. The radius and the diameter are not two independent measurements of a circle, they are the same measurement written two ways, so filling both is not more accurate — it is a contradiction. A radius of 5 and a diameter of 12 do not describe the same circle, and the page says so rather than quietly preferring one of them. Fill the box you actually measured and leave the other empty; if you filled the wrong one, clear it rather than adding the second. The units are the part most likely to catch you out. Either box accepts centimetres, metres, millimetres, inches or feet, and the answer is always in centimetres, because the conversion happens on the way in rather than on the way out. Enter a radius in inches expecting inches back and the figure will be about two and a half times too large. Nothing here is squared, so the conversion is a single division by 2.54 — but that is specific to this page, and the habit of dividing by 2.54 is a habit that will produce the wrong answer the moment you move to an area. One last thing before you check the arithmetic yourself. The answer is worked out from an unrounded radius and rounded only at the very end, which matters when you start from the diameter of a circle whose radius is not a round number. Work through a diameter of 12 by hand, keeping the radius as 6, and you will match the page exactly. Where people and the page part company is when the radius is a value that cannot be written down, which happens whenever the measurement you started with was the circumference itself.

The circumference of a circle from its radius, with the diameter alongside

Radius (cm)Diameter (cm)Circumference (cm)
126.2832
2412.5664
3618.8496
51031.4159
102062.8319
2040125.6637
0.513.1416
000

Eight circles and three columns, the last of which is the answer while the first two explain it. The diameter column is here because the results panel does not echo what you typed: if you entered a diameter, this is the only place to see what it was read as. The first row is the unit circle, whose circumference is numerically 2π, which is worth knowing on its own and is the quickest way to remember what the formula does when the radius is 1. The first two rows are the pair that shows the scaling rule: the radius goes from 1 to 2 and the circumference from 6.2832 to 12.5664, exactly double, because a circumference is a length rather than an area. The fourth row is the pair the page loads with, 31.4159, and it is the figure the overview page prints for the same circle. The seventh row is a radius of half a centimetre, whose circumference is π to four places, and the last is a radius of zero, where the circle has shrunk to a point and the answer is a real zero rather than a missing one.

Formula

C = 2πr

π
The constant every circle shares, about 3.14159. It is defined as a circle's circumference divided by its diameter, which is where it comes from and why dividing by it is how you go back the other way
r
The radius in centimetres — the distance from the centre out to the edge, or half the diameter if the diameter is what you measured
2πr
The formula: the radius doubled, then multiplied by π. Doubling first is arithmetically the same as starting from the diameter and multiplying by π once, and the page writes it that way so that both boxes land on the same detailed figure
Circumference
How far it is around the circle once, in centimetres, whatever unit the input box was set to
Diameter ÷ 2
The way in from the other box. Halving is exact whenever the diameter is a value you measured, which is why this page gets the same answer as the one that starts from a radius
Twice the radius
What doubling the radius does here: the circumference doubles too. That is not true of the area of the same circle, which goes up fourfold, and reaching for the wrong one of those two habits is the commonest arithmetic slip on round shapes
Four decimal places
How wide the reading is printed. A circumference is almost never a number that ends unless the radius was chosen to make it so; 2π times a radius of 5 is 31.41592... and the page shows 31.4159

Anything that has to go around something else. Fencing, edging and borders are the plainest case: the length of fence around a circular pen, the edging strip around a round lawn, the trim around a circular table top or a round mirror, all of them are this formula and all of them are bought by the metre. Belts, bands and hoops are the second: a drive belt around two pulleys of the same size, a metal hoop around a barrel, a gasket ring, a circular weatherstrip, the retaining band around a tank lid — each is cut to a circumference, and the difference between the inside and outside of the material is a separate allowance that this page does not make. Rolling is the third and the least obvious: a wheel of known diameter travels its circumference in one full turn, so the distance a bicycle, a car or a measuring wheel covers is the circumference multiplied by the number of revolutions, which is exactly how a mechanical odometer works. In building and landscaping it is the length of a circular kerb, the coping around a round pool, the amount of channel drain needed around a circular drive, and the run of a circular wall. In engineering it is the belt length around a single pulley, the circumference of a pipe that a wrap of insulation or a length of tape has to cover, and the travel per revolution of any roller. In the classroom it is the first place a pupil meets the idea that a constant can be irrational and still be useful: wrap a piece of string around a tin, straighten it against a ruler, divide by the diameter, and the answer comes out a little over three every time — which is the whole reason pi is worth writing down. And in everyday arithmetic it is the length of ribbon around a jar, the piping around a cushion, or the question of whether a round tablecloth is big enough to drop over the edge.

Worked examples

  1. A radius of 5

    1. Double the radius: 2 × 5 = 10
    2. Multiply by π: 10 × 3.14159 = 31.4159

    The pair the page loads with, and the figure that ties this page to two others: a circle of radius 5 measures 31.4159 around here, on the overview page, and on the ellipse page when its two axes are both 10. All three agree to the last digit because none of them rounds the radius before multiplying.

  2. A diameter of 12

    1. Halve the diameter to get the radius: 12 ÷ 2 = 6
    2. Double it again inside the formula: 2 × 6 = 12
    3. Multiply by π: 12 × 3.14159 = 37.6991

    The commonest way people arrive here, since measuring across a circular object is easier than finding its centre. The halving and the doubling cancel exactly, so the answer is the same figure the overview page prints for a diameter of 12 — the detour through the radius costs nothing and is what keeps the two pages in agreement.

  3. A radius of 10

    1. Double the radius: 2 × 10 = 20
    2. Multiply by π: 20 × 3.14159 = 62.8319

    Exactly twice the circumference of a circle of radius 5, and the cleanest place to see that the perimeter of a circle scales with its radius rather than with its square. Compare it with the area page, where the same change from 5 to 10 multiplies the answer by four. Two pages, one changed number, two different factors, and knowing which is which is most of what there is to know about round shapes.

  4. A radius of half a centimetre

    1. Double the radius: 2 × 0.5 = 1
    2. Multiply by π: 1 × 3.14159 = 3.1416

    A circle of radius one half has a circumference of exactly π, numerically, which is a tidy thing to notice and a quick way to remember what the formula does when r is a half. Small circles like this are where the four decimals stop being generous: at this size the reading is pi to four places, and any further digit would be noise on a measurement nobody can take that precisely.

  5. A radius of 0

    1. Double the radius: 2 × 0 = 0
    2. Multiply by π: 0 × 3.14159 = 0

    A circle with no size has nothing to go around, and zero is the right answer rather than a missing one. Zero is a legal value in either box — the difference between measuring zero and not measuring at all — and an empty box is the case the page treats as having nothing to work from.

Limitations

This page returns the circumference and nothing else: not the radius, not the diameter, not the area. If a circumference is what you already have and you want one of the others, that is a different page. The answer is always in centimetres whatever unit each box is set to, because the conversion happens on the way in; a radius entered in inches comes back as a centimetre figure that has to be divided by 2.54 to be read in inches, and that single division is right for this page only — areas and volumes convert differently. The answer is computed from an unrounded radius and rounded once at the end, so a hand calculation that rounds the radius first can differ in the last digit; the page is not the one that is wrong. Four decimal places is a display width rather than a claim of exactness — a circumference is usually an endless decimal, and 31.4159 is a rounded 31.41592... Exactly one box may be filled: give the page two, even two that agree with each other, and it refuses, because two boxes are two measurements and the page has no way to know which of them you meant. A circumference of zero is accepted and answered with zero. Nothing here handles an ellipse, a segment, a sector or a spiral — an ellipse's perimeter in particular has no elementary formula at all, so a page that offers one is offering an approximation and should say so.

Frequently asked questions

Why can I only fill in one of the two boxes?
Because the radius and the diameter are the same measurement written two ways, so any two figures are either redundant or contradictory. A radius of 5 and a diameter of 12 cannot both be true of one circle. The page refuses two rather than picking one, because the point of having both boxes is that you fill in whichever one you actually measured.
I can only measure straight across the circle. Which box is that?
The diameter, and it is the more common way in — a tape across a tin, a table top or a pipe is much easier than finding the centre and working outwards. The page halves it to a radius and then doubles it again inside the formula, which sounds like a detour and is not: it is what keeps the figure identical to the one the overview page prints for the same circle.
I entered inches. Why is the answer in centimetres?
Because the conversion happens on the way in, not on the way out. Your inches are turned into centimetres, the circumference is worked out, and the answer is in centimetres. To read it in inches, divide by 2.54. That single division is correct here because a circumference is a length — the same divisor would be wrong on an area, where the length factor is squared.
Why is the circumference of a circle of radius 1 not exactly 6.2832?
Because it is exactly 2π, and 2π is an endless decimal — 6.283185... The page prints it to four decimal places, so what you are seeing is 6.2832 rounded. The reading is a display width, not a claim that the number terminates, and no radius other than zero and a few specially chosen values will give a circumference that ends.
Does doubling the radius double the circumference?
Yes, and it is the one place where an instinct carried over from the area formula will lead you wrong. A circumference is a length, so it grows in proportion to the radius; the area of the same circle grows with the square of the radius, so doubling that goes up fourfold. The table shows the first two rows for exactly this reason: the radius goes from 1 to 2 and the circumference from 6.2832 to 12.5664.
Can I get the radius or the diameter back from a circumference?
Not on this page — it takes a radius or a diameter and gives the circumference, one direction only. Going back is a division by π, and the page that does it in both directions, printing the diameter and the circumference together, is the circumference to diameter page. The overview page will also give you all four quantities from any one of them.

References

Related calculators

Circle CalculatorRadius, diameter, circumference and area from any one of them.Circumference to Diameter CalculatorDiameter from a circumference, and the reverse.Circle Area CalculatorHow much room there is inside a circle, in square units.Arc Length CalculatorThe length of the arc between two points on a circle.