Circle Calculator

Pi times the radius squared, for common radii

"Pi times r squared" is just the area formula said out loud. Square the radius first, then multiply by pi (about 3.14159). So a radius of 13.5 gives 13.5 times 13.5, which is 182.25, and 182.25 times pi is about 572.56 square units. The circumference is the distance around the edge, which is two times pi times the radius. Any radius works in the calculator above; these are the ones people look up most.

Rounded to two decimal places. Results are in the same units you measure the radius in, so a radius in centimetres gives an area in square centimetres. If you have the diameter rather than the radius, halve it first.

Pi times a number, and the circle it measures

Multiplying by pi without squaring anything first is a different calculation from the area one above. Pi times a number is the circumference of a circle that many units wide, because circumference equals pi times the diameter. So 47 times pi is about 147.65, and that is the distance once around a circle 47 units across. This is also what people mean by the length of a circle. Squaring first would give the area instead, which for a radius of 47 is a far larger 6,939.78.

Rounded to two decimal places. Pi times n answers the number on its own, and the last two columns read n as a diameter, which is the circle that number describes. Half of any area in the last column is the matching semicircle, so a circle 10 wide covers 78.54 square units and its half covers 39.27.

When your number is already squared

If you squared the radius yourself before reaching for a calculator, the number you are holding is r squared, not the radius and not the diameter. Multiplying it by pi finishes the area calculation. So 56.25 times pi is 176.71, and that is the area of a circle of radius 7.5, because 7.5 times 7.5 is 56.25. The quickest way to tell which table you need: if your number has an obvious square root, it is probably r squared and this table is the right one.

Rounded to two decimal places, radii to four. The same number read as a diameter instead gives a much smaller circle, which is the table above this one.

Working backwards: area or circumference to radius

Both formulas run in reverse, and dividing by pi is the first step of each. Divide a circumference by pi and you have the diameter in one move, so 13.5 divided by pi is 4.30 and that is the width of a circle 13.5 units around. Divide an area by pi and you have the radius squared, which still needs a square root: 78.54 divided by pi is 25, and the square root of 25 is a radius of 5. The calculator above takes an area or a circumference as its input for exactly this, so any value not in these tables can go straight into it.

If you know the area

If you know the circumference

Areas and circumferences here are rounded inputs, so a radius reads as 4.9999 rather than a flat 5 in places. That is the rounding in the number you started from, not an error in the formula.

Half circles, quarter circles and arcs

A part circle needs no new formula. Work out the full circle and take the fraction you want, so a semicircle is the area divided by two and a quarter circle is the area divided by four. A circle of radius 5 covers 78.54 square units, which makes the semicircle 39.27 and the quarter 19.63. The one place people slip is the perimeter: the curved edge of a semicircle is half the circumference, but the shape is only closed once you add the straight diameter across the bottom, so 15.71 of curve plus 10 of straight edge gives a perimeter of 25.71.

Rounded to two decimal places. Doubling works the same way in reverse: a semicircle of 28.27 square units belongs to a full circle of 56.55.

Reading the notation

The same two formulas get written half a dozen ways, and the shorthand is what usually causes the mix-up rather than the arithmetic. Every form in the left column is one of the two calculations this page does, and the thing to watch is whether the radius is squared. Squaring gives an area, in units squared. Not squaring gives a distance around the edge, in plain units.

Pi is an endless decimal, so every result here is cut to two places. The calculator above carries the full precision and rounds only what it shows.