Area Of Circle Calculator

What this calculator does

You have got something round. A tabletop, a garden plot, a pipe, a pizza. And you need the space inside it.

Measure the radius, the distance from the centre of the circle straight out to the edge, type it in, pick your unit, and the tool gives you the area. That is the whole job. One number in, one number out.

So if you know how far it is from the middle to the rim, you are ready.

Using the calculator

1. Type the radius into the box.
2. Pick the unit it is measured in (millimetres, centimetres, metres, kilometres, inches, feet, yards or miles).
3. Hit Calculate Area.

The answer comes back in square units of whatever you chose. Radius in centimetres gives you area in square centimetres. The radius has to be a positive number, since a circle cannot have a negative width.

One handy thing. If you type a radius and then switch the unit, the tool converts that radius into the new unit for you, so it stays the same real length. For the tidiest answer, pick your unit first and then type the number. The reason why is in the precision part below.

The formula | area = π × radius²

The area of a circle is:

area = π × radius²

Take the radius, multiply it by itself, then multiply by π. That is all there is to it.

But what is π actually? π (pi) is the fixed number that links a circle's size to the space inside it. It is the same for every circle, big or small, and it sits at roughly 3.14159. Its digits run on forever and never settle into a repeating pattern, which is the short way of saying π is an irrational number. Johann Lambert proved that back in 1768, and no one has ever found a last digit since.

So why does squaring the radius work? Picture slicing the circle into a load of thin wedges, like a pizza, then laying them out in a row, tip up, tip down, tip up. They line up into something close to a rectangle, as tall as the radius and as long as half the way around the circle. The thinner you slice, the closer it gets to a perfect rectangle. Archimedes worked this out more than two thousand years ago in his Measurement of a Circle, and it lands on the same place we started: area = π × radius².

Under the hood, the calculator uses your browser's own built-in value of π, which is good to about 15 to 16 digits, and it runs the whole sum right here on your device. Nothing is sent anywhere, and there is no heavy maths library loaded behind it. Just π and a multiplication.

A worked example with a 5 cm radius

Say the radius is 5 cm. Here is the whole thing by hand.

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

So the area is about 78.54 square centimetres. The tool rounds that long decimal and shows you 78.540.

If you want the exact value instead of a decimal, just leave π in place: the area is 25π cm². That is the same number, written without rounding anything off.

Getting the units right

So why square units? Because you are multiplying a length by a length. A radius of 5 cm times itself is 5 cm × 5 cm = 25 cm², not 25 cm. The unit gets squared right along with the number.

Forgetting to square the unit is the most common slip with circle area, so the tool labels the result for you: square mm, square cm, square m, and so on, matching whatever you measured the radius in.

The unit menu does one more job. Change the unit after typing a radius and the tool converts that radius using standard conversion factors, so it keeps pointing at the same real length. That is convenient, but the conversion also re-tidies the number a little. So if you care about the last digit, pick your unit first, then type the radius, then read the area.

How precise the answer is

Two things decide how precise your result is.

First, π. The tool uses the browser's built-in value, accurate to about 15 to 16 significant digits. That is far finer than anything you can measure with a ruler or a tape, so π is never the weak link in your answer.

Second, the way the number is shown. A short, tidy result is shown in full. A long, messy decimal is rounded to three decimal places so it stays readable. And a very small area, one that would otherwise round away to plain zero, is shown to ten places instead, so you still get a real number to work with.

Here is the part worth holding onto. Your area is only ever as good as the radius you put in. If you measured the radius to two solid digits, the area is good to about two digits too, however many the screen shows you. So once you have the result, round it back to match how well you measured. The extra digits are real arithmetic, but they are not extra accuracy.

If you only have the diameter or circumference

The tool asks for the radius, but you can get there in one small step.

Measured the full width straight across the middle? That is the diameter, and the radius is just half of it:

radius = diameter ÷ 2

Measured around the outside instead? That is the circumference, and you can get the radius from it like this:

radius = circumference ÷ (2 × π)

Work out the radius either way, type it in, and you are back on track. If you would rather the tool did the rest of the circle for you, the circle calculator takes any one of radius, diameter, circumference or area and fills in the others.

Area of a circle for common radii

A few common radii worked out, both as an exact value and as the tool rounds it. The areas are in square units of whatever unit you measured the radius in.

Questions people ask

What is the area of a circle with a radius of 5?

It is 25π, which works out to about 78.54 square units. Square the radius (5 × 5 = 25), then multiply by π.

Is the formula πr² or πd²?

It is π × radius². If all you have is the diameter, halve it first to get the radius (radius = diameter ÷ 2), then square that. Plugging the diameter straight into π × d² gives an answer four times too big.

How do I find the area from the diameter?

Halve the diameter to get the radius, then use the formula. So radius = diameter ÷ 2, and area = π × radius².

Why is the area in square units?

Because area is a length multiplied by a length. Centimetres times centimetres give square centimetres. The unit gets squared together with the number, which is why a radius in cm produces an area in cm².

How many digits of π does this use?

About 15 to 16, the value built into your browser. The calculation itself runs at that precision, and the answer on screen is then rounded so it stays easy to read.

Can I get the answer in terms of π?

The tool gives you the decimal. For the exact form, leave π in place, for example 25π for a radius of 5. That is the same value with nothing rounded off.

References

A quick note on where this comes from. The formula itself is old. Archimedes proved that a circle's area equals π × radius² more than two thousand years ago, in his Measurement of a Circle. That π never ends and never repeats was settled by Johann Lambert in 1768. And the value of π the tool leans on, along with its properties, is the one tabulated by the US National Institute of Standards and Technology. For further reading, see Area of a circle.

1. Archimedes, Measurement of a Circle (c. 250 BCE), the classical proof that a circle's area equals π × radius².
2. J. H. Lambert (1768), the first proof that π is an irrational number.
3. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value and properties of π. https://dlmf.nist.gov/