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CalcMax

Circle Area 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

Result

78.5398 cm²

Area

A circle area calculator takes whichever of the radius, the diameter or the circumference you happen to have measured and returns how much surface the circle encloses, in square centimetres. The formula is π times the radius squared — pi r squared, as it is usually said — and the middle step — turning whatever you typed into a radius — is where the three boxes come from. A diameter is a radius doubled and a circumference is a radius multiplied by 2π, so any one of the three determines the other two, which is why you fill exactly one. Filling two 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 picking one of them. The word for what is being measured is worth a moment. Strictly the area belongs to the disk — the flat region — while the circle is the curve around its edge. Nobody says it that way in ordinary use, and this page follows ordinary use, but it is the reason a formula that looks like it should give a length comes back in squared units. The units are the part of this page most likely to catch you out. The radius can be entered in centimetres, metres, millimetres, inches or feet, and the answer is in square centimetres whatever you choose, because the conversion happens on the way in rather than on the way out. Enter a radius in inches expecting square inches and you will get a number roughly six and a half times too large. The rule to carry away is that areas convert by the square of the length factor, not by the factor itself. There is one more thing worth knowing before you compare this page against your own arithmetic. The area is worked out from the unrounded radius and rounded only at the end, which matters when you start from a circumference, because then the radius is not a number anyone can write down. Type a circumference of 20, read off the radius the table gives you, square that and multiply by π, and you will land one digit away from the answer this page prints. The page is right and so is your arithmetic; the difference is that one of you rounded in the middle and the other did not.

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

Radius (cm)Diameter (cm)Area (cm²)
123.1416
2412.5664
3628.2743
51078.5398
1020314.1593
20401256.6371
0.510.7854
000

Eight circles and three columns, and the third column is the answer while the first two explain it. The radius and its diameter are together because this page does not echo what you typed: if you entered a diameter, the only place to see what it was read as is here. The first row is the unit circle, whose area in square centimetres is numerically π, which is a neat thing to know and a quick way to remember what πr² does when r is 1. The first and second rows are the pair that shows the relationship worth remembering: the radius goes from 1 to 2 and the area from 3.1416 to 12.5664, four times as much. The fourth row is the pair the page loads with, and it is the row the overview page also prints for the same circle, 78.5398 — the two pages agree on that figure to the last digit because neither of them rounds the radius before squaring it. The seventh is a radius of half a centimetre, whose area is a quarter of π, and the last is a radius of zero, where the circle has shrunk to a point and the area is a real zero rather than a missing answer.

Formula

A = πr²

π
The same constant for every circle, about 3.14159. It is the ratio of a circle's circumference to its diameter, which is where it comes from and why it turns up in both the length formulas and this one
r
The radius, in centimetres — half the diameter, or the circumference divided by 2π, depending on which box you filled
r²
The radius multiplied by itself, not by two. Doubling the radius multiplies the area by four, which is the single most useful fact about this formula and the reason the table has a pair of rows to show it
Area
How much flat surface the disk encloses, in square centimetres, whatever unit the input was entered in
Circumference ÷ 2π
How a radius is recovered from a measurement taken round the edge. The result is not a number that ends, so it is kept unrounded until after the squaring
Diameter ÷ 2
The other way in, and the easier one: the radius is exactly half the diameter whenever the diameter is a number you measured
Four decimal places
How wide the reading is written. The area of a circle is almost never a number that ends, so the four decimals are a display width rather than a claim that the figure is exact

Anything round whose surface is being covered, coated, cut or priced. Painting and coating is the plainest case: a circular table top, a tank lid, a round sign, a disc of metal being sprayed, all of them are quoted per unit area and the area is this formula. Materials cut from sheet are the second: a round blank stamped out of steel, a circular pane of glass, a disc of gasket, a round tablecloth — and there the area is what tells you how much of the sheet each one consumes, which is the number the price depends on. Landscaping and building use it for round beds, circular patios, round lawns, the cross-section of a column, and the surface of a circular pond. In engineering it is the cross-sectional area of a pipe, a shaft or a cable, which is what flow rate and current-carrying capacity are reckoned against — a pipe of twice the diameter carries four times the area, and that factor of four catches people out in both directions. In physics and in everyday arithmetic it is the area that turns a pressure into a force, an intensity into a total, or a rainfall figure into a volume of water. And there is the classroom use, which is the one that makes the formula stick: draw a circle on squared paper, count the whole squares inside it, and compare the count with πr². The count comes out a little low, because the squares at the edge are partly outside, and that gap is the first honest encounter most people have with what an irrational constant in a formula actually means.

Worked examples

  1. A radius of 5

    1. Square the radius: 5 × 5 = 25
    2. Multiply by π: 25 × 3.14159 = 78.5398

    The pair the page loads with, and the row that ties this page to the overview page: a circle of radius 5 has an area of 78.5398 there too. Nothing was rounded in between, which is why the two agree to the last digit — an area worked out from a radius that had already been rounded would not.

  2. A diameter of 12

    1. Halve the diameter to get the radius: 12 ÷ 2 = 6
    2. Square it: 6 × 6 = 36
    3. Multiply by π: 36 × 3.14159 = 113.0973

    The commonest way people arrive here, because a circular object is much easier to measure across than from the centre to the edge. This page does not print the radius back at you, so if you want to check that your 12 was read as 6, look at the radius column of the table — 6 is not on it, but the row for 3 has a diameter of 6 and the row for 10 has a diameter of 20, which is enough to see the relationship the page is using.

  3. A circumference of 20

    1. Recover the radius: 20 ÷ (2 × π) = 3.1830988…
    2. Square it: 3.1830988 × 3.1830988 = 10.1318311…
    3. Multiply by π: 10.1318311 × 3.14159 = 31.8310

    The row that explains why this page never rounds in the middle. The radius here cannot be written down — it is 10/π — so it is carried unrounded through the squaring and only the answer is rounded. Do it the other way: read the radius off the table as 3.1831, square that, multiply by π, and you get 31.8309. One digit out, and neither working is wrong: one of them rounded before the end.

  4. A radius of 10

    1. Square the radius: 10 × 10 = 100
    2. Multiply by π: 100 × 3.14159 = 314.1593

    Exactly four times the area of a circle of radius 5, and the cleanest place to see that doubling the radius quadruples the area. The radius went up by a factor of two and the answer by a factor of four, because the radius is squared. It is the same relationship that makes a pipe of twice the diameter carry four times as much.

  5. A radius of 0

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

    A circle of no size encloses nothing, and the answer of zero is correct rather than missing. Zero is a legal value in any of the three boxes — it is the difference between a measurement of zero and no measurement at all, and the page treats the second of those, an empty box, as the case with nothing to work from.

Limitations

This page returns the area and nothing else: not the circumference, not the diameter, not the radius. Given an area and wanting one of the others is the other page. The output is always in square centimetres, whatever unit each input is set to — the conversion happens on the way in, so a radius entered in inches comes back as a square-centimetre figure that has to be divided by 6.4516 to be read in square inches. Converting an area by dividing by the length factor of 2.54 rather than by its square is the error to watch for. The answer is worked out from an unrounded radius, so if you start from a circumference and check the arithmetic by hand using a rounded radius, your figure will differ in the last digit; the page is not the one that is wrong. Four decimal places is a display width and not a claim that the area is exact — an area of a circle is almost never a number that ends, and 78.5398 is a rounded 78.53981633... A circle of radius zero is accepted and gives an area of zero, which is correct. This page takes exactly one of the three measurements: give it two and it refuses, because two inconsistent figures do not describe a circle. Nothing here handles an ellipse, a segment, a sector, or the surface of a sphere — each of those is a different shape with a different formula, and an ellipse in particular is not a stretched circle with an easy area.

Frequently asked questions

Why can I only fill one of the three boxes?
Because the three describe the same circle, so any two of them are redundant and any two that disagree are a contradiction. A radius of 5 and a diameter of 12 cannot both be true of one circle. The page refuses two rather than quietly choosing one, because the whole point of the boxes is that you enter whichever measurement you actually have.
I entered inches. Why is the answer in square centimetres?
Because the unit conversion happens on the way in, not on the way out. Your inches are turned into centimetres, the area is computed, and the answer is in square centimetres — it is not a square-inch figure. To read it in square inches, divide by 6.4516, which is 2.54 squared. Areas convert by the square of the length factor, and dividing by 2.54 instead is the mistake to avoid.
Why do I get a slightly different answer when I check it by hand?
Because of where the rounding happens. The page keeps the radius unrounded until the very end, and if you started from a circumference then the radius is a number that cannot be written down exactly. Read it off the table, square it and multiply, and you are rounding in the middle. That is worth 0.0001 on a circumference of 20, and this page is the one that agrees with itself.
Why did doubling the radius give four times the area?
Because the radius is squared rather than doubled. Twice the radius means twice the radius in each of the two directions the square is made of, so the area goes up by two times two. It is worth holding on to: it is the same relationship that makes a pipe of twice the diameter carry four times the flow, and it catches people out in both directions.
Does the page show me what it read my diameter as?
Not directly — the results panel shows the area only, and does not echo what you typed. The table is there for exactly that reason: every row gives a radius next to its diameter and its area, so you can find a row near your figure and see the doubling the page is applying. A diameter of 12 is a radius of 6, which is the row for 3 on the table with its diameter doubled.
Can I work out the radius from an area here?
Not on this page. It is the other way round deliberately: this page takes a length and gives an area, and going back the other way means a square root. There is a page that does it along with everything else, taking any one of the four quantities and returning the rest, and that is the one to use if the area is what you started with.

References

Related calculators

Circle CalculatorRadius, diameter, circumference and area from any one of them.Circumference CalculatorCircumference of a circle from its radius or diameter.Circumference to Diameter CalculatorDiameter from a circumference, and the reverse.Sector Area CalculatorThe slice of a circle cut by two radii: area, arc length and chord from the angle.