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CalcMax

Circle Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm² – 1,000,000,000,000,000,000 cm²

Result

5.0000 cm

Radius

Diameter
10.0000 cm
Circumference
31.4159 cm
Area
78.5398 cm²

A circle calculator takes whichever one of four measurements you happen to know and gives you the other three. Radius, diameter, circumference and area are all the same circle described four ways, so any one of them pins down the rest: type 5 into the radius box and the page returns a diameter of 10, a circumference of 31.4159 and an area of 78.5398. The arithmetic behind that is two formulas and their inverses. Circumference is π times the diameter, area is π times the radius squared, and dividing by two, by two π or by π gets you back to the radius from any of the other three. This is the reason the page asks you to fill in exactly one box rather than making the radius compulsory: filling in the radius would be fine if the radius were always what you had, but a circle drawn on a plan often comes to you as its area or as a length of edging, and a page that insisted on the radius would make you do the division yourself before it would help. So all four boxes are optional and any one of them will do. Filling two is refused rather than ignored, because two of the four are never independent — a radius and a diameter that disagree have no circle behind them, and quietly picking one would hide the disagreement rather than show it. Two things about the answers are worth knowing before you use them. The results are in centimetres and square centimetres, whatever unit the dropdown beside the input was left on. That is a deliberate choice rather than an oversight: the four outputs are a fixed set computed the same way for everyone, the reference table below is computed in the same unit so that the page does not contradict itself, and switching the input to inches only changes how your number is read on the way in. If you measured in inches, the answer is still shown in centimetres and you will need to convert it back. And the four printed numbers belong to the value you typed rather than to each other. A circumference of 20 gives a radius of 3.1831, and the true area of that circle is 31.8310; multiplying the printed radius by itself and by π gives 31.8309 instead, because 3.1831 is already a rounded version of a number that never ends. The area printed is the correct one — it comes from the unrounded radius — and the discrepancy is a property of writing an irrational number down.

Circles reached from each of the four starting points

GivenRadius (cm)Diameter (cm)Circumference (cm)Area (cm²)
r51031.415978.5398
d61237.6991113.0973
C3.18316.36622031.831
A5.641911.283835.4491100
r126.28323.1416
r0.513.14160.7854
r0000
A564.18961128.37923544.90771000000

Eight circles, and the first column says which of the four values each row starts from — r for a radius, d for a diameter, C for a circumference and A for an area. That column is there because all four numbers are printed on every row: once the circle is solved there is nothing missing from it, so without the first column you could not tell which of the four was the one that was given. The first four rows take the four starting points in order, and the first of them is the input the page loads with. The third row is the one to study, since it starts from a circumference and shows the arithmetic that makes the printed numbers fail to reproduce each other: the area of 31.831 is a hundred divided by π, and squaring the printed radius would give 31.8309 instead. The fourth row starts from an area of a hundred and gets a radius of 5.6419, which is the square root people forget — dividing a hundred by π would have given 31.831. The fifth and sixth rows are the unit circle and its half, and between them they make a point worth a second look: the area of the circle of radius one is 3.1416, which is exactly the circumference of the circle of radius a half, printed one row below it. The same four digits mean an area in one row and a length in the other, which is the sharpest reminder on this table that a column heading carries as much of the answer as the number does. The seventh row is a circle of radius zero, where every column is zero and the answer is real rather than empty. The eighth starts from an area of a million square centimetres and gives a radius of 564.1896 centimetres, a little over five and a half metres, which is the sort of input the area box is scaled for. Note also that the first column is not evenly divided: four of the eight rows start from a radius, because the radius is the value the other three are defined from, while the diameter and the circumference appear once each. Every number here is recomputed from its starting value when the page is built, and all of them are in centimetres.

Formula

d = 2r C = 2πr = πd A = πr² and back: r = d ÷ 2 = C ÷ 2π = √(A ÷ π)

Radius
The distance from the centre to the edge, in centimetres. Zero is allowed and describes a circle of no size, in which case all four outputs are zero
Diameter
The distance right across through the centre, twice the radius. Filling this box instead of the radius changes nothing about the answer, since the page halves it first thing
Circumference
The distance all the way around, π times the diameter. This is the one input that requires a division by two π to get back to the radius, so it is where the four printed digits stop being enough to reproduce the others by hand
Area
The space inside the circle, π times the radius squared, in square centimetres. Going the other way means dividing by π and taking a square root, which is why the area box accepts values up to a much larger number than the three length boxes
π
The ratio of a circle's circumference to its diameter, about 3.14159. The page uses the full double-precision value rather than shortening it to three decimal places, so a hand calculation done with 3.14 will drift from the printed answer
Four decimal places
How wide every output is written. The radius and diameter are often exact and print with trailing zeros; the circumference and the area are almost always irrational, so for those two the four digits are where the answer is cut rather than where it ends

The page is useful precisely because a circle rarely arrives as a radius. A round table is sold by its diameter, a length of fencing around a circular bed is sold by the metre and is a circumference, a tin of paint covers so many square metres and the patch you are painting is a circle, and a bicycle wheel is quoted by the size of its tyre. Each of those comes with a different one of the four numbers and each of them can go straight into the corresponding box. The reverse direction is where people go wrong by hand. Given the area, the radius is not the area divided by π — it is the square root of that, and forgetting the root produces a radius that is out by a factor of the square root of itself, which for a mid-sized garden is not a small error. Given the circumference, the radius is the circumference divided by two π, and using 3.14 instead of the full value is enough to move the last printed digit. The page does both correctly, which makes it a reasonable way to check a hand calculation you are not sure of. One habit is worth carrying over from the geometry around this page: whatever you compute, the answer should be roughly a third of the circumference and a sixth of the area's square root, so if a diameter comes back larger than the circumference you have put the input in the wrong box.

Worked examples

  1. A radius of five

    1. Diameter: 5 × 2 = 10
    2. Circumference: 2 × π × 5 = 31.415926…, which rounds to 31.4159
    3. Area: π × 5 × 5 = 78.539816…, which rounds to 78.5398
    4. The radius comes back unchanged, since it is what was given

    The input the page loads with, and the easiest to check: a circle of radius five has a diameter of ten and an area a little over three quarters of a hundred, which is about right since π is a bit over three and five squared is twenty-five. The radius is printed as 5.0000 rather than 5, which is a display width and not a claim about precision — it is exactly five.

  2. A diameter of twelve

    1. Radius: 12 ÷ 2 = 6
    2. Circumference: π × 12 = 37.699111…, which rounds to 37.6991
    3. Area: π × 6 × 6 = 113.097335…, which rounds to 113.0973
    4. The diameter stays as given

    The same shape reached from the other end, and a reminder that the diameter and radius outputs are exact while the other two are not — dividing an even number by two introduces no error at all, while multiplying by π always does. The area being a little over 113 is a reasonable check: six squared is thirty-six and thirty-six times π is a bit more than a hundred and thirteen.

  3. A circumference of twenty

    1. Radius: 20 ÷ (2 × π) = 3.1830988…, which rounds to 3.1831
    2. Diameter: that radius doubled is 6.3661977…, which rounds to 6.3662
    3. Area: π × the unrounded radius squared, which is 100 ÷ π = 31.830988…, and rounds to 31.8310
    4. The circumference stays as given

    The case that shows why the four printed numbers do not always reproduce each other. The area is exactly one hundred divided by π, and it is computed from the unrounded radius rather than from the 3.1831 on screen; squaring the printed radius and multiplying by π gives 31.8309 instead. The printed figure is the correct one. Reading it the other way round, the area of a circle of circumference 20 is a third of a hundred, which is the quick mental check that says 31.83 is in the right region.

  4. An area of one hundred square centimetres

    1. Radius: √(100 ÷ π) = 5.6418958…, which rounds to 5.6419
    2. Diameter: that doubled is 11.2837916…, which rounds to 11.2838
    3. Circumference: 2 × π × the unrounded radius = 35.449077…, which rounds to 35.4491
    4. The area stays as given

    The direction people get wrong on paper: the radius is the square root of the area divided by π and not the area divided by π, which would give 31.831. A hundred square centimetres is a shape about eleven centimetres across, which is a useful sanity check — a circle of radius 31.8 would have an area of three thousand, not one hundred. That same number 31.831 is the area computed in the example above, from a circumference of twenty; here it is what the wrong calculation would produce, which is a coincidence of these two inputs rather than anything to read into.

  5. A radius of zero

    1. Diameter: 0 × 2 = 0
    2. Circumference: 2 × π × 0 = 0
    3. Area: π × 0 × 0 = 0
    4. All four outputs are zero, which is the correct reading of a circle with no size

    Zero is a legal input here rather than an empty box. A circle of radius zero is a single point, and every one of the four measurements is zero, so a row of zeros is a real answer. The page distinguishes this from leaving everything blank, which is refused with a message asking you to fill one box — the difference between a shape that has collapsed to nothing and a form that has not been filled in.

Limitations

The outputs are always centimetres and square centimetres, whatever unit the input dropdown is set to. Choosing inches in the dropdown changes how your number is interpreted on the way in and nothing about how the answer is shown, so a circle entered as ten inches comes back with a radius of 25.4 centimetres. That is a deliberate choice and the reference table follows the same rule so the page never contradicts itself, but it does mean you have to convert the answer yourself if you wanted it in the unit you typed. The four printed values are each derived from the value you entered rather than from one another, so they do not always reproduce each other to the last digit: multiplying the printed radius by π can differ from the printed circumference in the fourth decimal place, and the printed circumference is the one to trust. Circumference and area are almost always irrational numbers, so four decimal places is a cut rather than a complete answer, and rounding twice — reading a printed value and using it in the next calculation — is how the discrepancy appears in the first place. The radius and diameter are exact whenever the input was, but they are rounded too when they come from an area or a circumference. At most one box may be filled: two or more are refused, because the four values are not independent and a pair that disagrees does not describe any circle. The page will not accept a negative value in any box, and it will not tell you that a circle of a given area has a radius you would have found faster by taking a square root rather than dividing.

Frequently asked questions

Which boxes do I fill in?
Exactly one, whichever one you happen to know. All four are optional, and any of them will do: the page takes the value you give it, works out the radius, and derives the other three from there. Filling two boxes is refused rather than resolved, because the four measurements are not independent of each other — a radius of 5 and a diameter of 12 cannot both describe the same circle, and the page would rather say so than silently pick one of them.
Why is the answer in centimetres when I typed in inches?
Because the four outputs are a fixed set in centimetres and square centimetres, chosen so that the results panel and the reference table below it always agree. The dropdown changes how your number is understood on the way in, not how the answer is written out, so a radius entered as 10 inches comes back as 25.4 centimetres. If you need the answer in inches, divide the length outputs by 2.54 and the area output by 6.4516.
Why does the area not match the radius that is shown?
Because the printed radius is already rounded and the area is not computed from it. The page works out the area from the full-precision radius and only rounds the display at the end, so for a circumference of 20 the area is 31.8310, while squaring the printed radius of 3.1831 and multiplying by π gives 31.8309. The printed area is the correct one. The same thing happens in reverse whenever you take a number off the screen and put it back into a calculation.
Does the page use 3.14 for pi?
No. It uses the full precision available in double-precision arithmetic, which is π to about fifteen significant digits, and rounds only the displayed result. That matters at the fourth decimal place: a circumference worked out by hand with 3.14 will usually differ from the printed answer in the last digit or two, and the difference is the shortened constant rather than an error in the calculation. For quick mental checks 3.14 is fine, and it is what most people should use.
How do I get the radius from the area?
Divide the area by π and take the square root. The square root is the step that gets dropped: without it you get a number the size of the area itself, which for a hundred square centimetres is 31.831 rather than the correct 5.6419. Sanity-check the result by squaring it and multiplying by π — you should land back on the area you started with, up to the rounding of the printed digits.
Can the radius be zero?
Yes, and all four outputs come back as zero, which is the honest description of a circle with no size. Zero is treated as a real input rather than as an empty box, so a row of zeros on the results panel is not a sign that something failed to fill in. A completely empty form is different: if none of the four boxes has a value the page asks for one, since there is nothing to work from.

References

Related calculators

Circumference CalculatorCircumference of a circle from its radius or diameter.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.Sphere CalculatorA ball: any one of radius, diameter or circumference gives volume and area.