Circle Calculator

What this calculator answers

Give it any one measurement of a circle, whether the radius, the diameter, the circumference or the area, and it calculates the other three. Each answer is shown exactly in terms of π (a 9-inch circle has an area of 20.25π in²) and as a decimal to the number of places you choose, with the steps, a drawing, the same circle in inches, feet, millimeters, centimeters and meters, and optionally the arc, sector and chord for an angle.

How to use it

• Radius, Diameter, Circumference, Area: type one measurement in its own box. Typing in a box clears the other three, and the line under each empty box shows its value calculated from the one you gave. You can type 1,250, 0.75 or 2.5e-3; fractions such as 3 1/2 need to be typed as decimals (3.5).
• Length unit: the unit your measurement is in. Every length in the answer uses the same unit, and the area uses its square, so inches give square inches (in²).
• Decimal places: how far the decimals are rounded, from 0 (whole numbers) to 10. Homework that says “to the nearest hundredth” means 2.
• Value of π: keep the full value unless a textbook or teacher asks for 3.14 or 22/7. The answers written with π stay exact either way; only the decimals change.
• Central angle (optional): an angle in degrees at the center of the circle, for the length of an arc, the area of a slice (a sector) and the straight chord across it.

The Try buttons load the cases worked through on this page: a 20-inch tape measurement around a post, a 7 cm radius with π as 22/7, a circle with an area of 1 m², and a 60° slice of the cake pan below.

Circle formulas: radius, diameter, circumference and area

$$d = 2r \qquad C = 2\pi r = \pi d \qquad A = \pi r^2$$

• $r$ is the radius, from the center to the edge.
• $d$ is the diameter, from edge to edge through the center: twice the radius.
• $C$ is the circumference, the distance around.
• $A$ is the area inside, in square units.
• $\pi$ is the ratio of any circle’s circumference to its diameter, about 3.14159.

Rearranged, each measurement gives the other three:

So a circle with a radius of 1 m has a circumference of 2π ≈ 6.2832 m and an area of π ≈ 3.1416 m².

Worked example: a 9-inch round cake pan

A recipe asks for an 8-inch square pan, and the pan you have is round and 9 inches across. Does it hold about the same amount of batter?

1. Radius: r = 9 ÷ 2 = 4.5 in.
2. Area: A = π × 4.5² = 20.25π ≈ 63.6173 in².
3. The square pan: 8 × 8 = 64 in². The round pan’s base is 0.6% smaller, so filled to the same depth it holds almost the same batter.
4. Circumference: C = π × 9 = 9π ≈ 28.2743 in, about 28 1/4 in around the rim.

The same pan in metric is 22.86 cm across with a base of 410.43 cm². An 8-inch round pan, by comparison, has only 16π ≈ 50.27 in² of base: the 9-inch pan is 12.5% wider but has 27% more area, because area grows with the square of the diameter (81 ÷ 64 ≈ 1.27).

How to find the radius from the circumference

Divide the circumference by 2π: r = C ÷ 2π. A post that measures 20 inches around has a radius of 20 ÷ 2π = 10/π ≈ 3.1831 in. The radius is not half the circumference; it is a little under one sixth of it (1 ÷ 2π ≈ 0.159).

How to find the diameter from the circumference (measuring a pipe, post or tree)

Divide the circumference by π: d = C ÷ π. Wrap a flexible tape or a piece of string once around the pipe, post or trunk, snug and square to its length, and read the circumference. For the 20-inch post, d = 20 ÷ π = 20/π ≈ 6.3662 in, which is 6 3/8 in to the nearest sixteenth. With the unit set to inches or feet, the calculator shows that fraction next to each length.

Measuring around is also the more accurate way to get a diameter. A tape reading that is off by 1/8 in moves the diameter by only 1/8 ÷ π ≈ 0.0398 in, while the same 1/8 in error measuring straight across goes into the diameter in full. The other way round, an error in a measured diameter is multiplied by π, about 3, in the circumference.

How to find the area from the circumference

Square the circumference and divide by 4π: A = C² ÷ 4π. For the 20-inch post, A = 20² ÷ 4π = 100/π ≈ 31.831 in². You get the same result by finding the radius first (10/π in) and using A = πr².

How to find the radius from the area

Divide the area by π and take the square root: r = √(A ÷ π). A circle with an area of 1 m² has a radius of √(1 ÷ π) = 1/√π ≈ 0.5642 m, a diameter of about 1.1284 m and a circumference of 2√π ≈ 3.5449 m. When the area is a whole number, the calculator also takes square factors out of the root, so an area of 50 gives a radius of 5√(2/π).

Answers in terms of π vs decimals

The decimal of π never ends or repeats, so any decimal answer is rounded somewhere. Writing the answer with π keeps it exact: the cake pan’s area is exactly 20.25π in², and 63.6173 in² is that value rounded to 4 decimal places. Answers in terms of π are what many algebra and geometry courses ask for; the decimal is what you need to cut, buy or build.

Some textbooks ask you to use 3.14 or 22/7 for π instead, and then their answer keys won’t match a calculator that uses the full value. Choose the same value in Value of π and the decimals will match. For a 7 cm radius:

In terms of π the answers are 14π cm and 49π cm² whichever value you choose. 22/7 gives round numbers here because 7 cancels; that is why textbooks that use it pick radii such as 7, 14 and 21.

Arc length and sector area

For a central angle θ in degrees, the arc and the slice are that share of the whole circle:

$$s = \frac{\theta}{360^\circ} \times 2\pi r \qquad \text{sector area} = \frac{\theta}{360^\circ} \times \pi r^2 \qquad c = 2r\,\sin\frac{\theta}{2}$$

The chord $c$ is the straight line between the ends of the arc: splitting the slice’s triangle down the middle gives two right triangles with half the chord, $r\,\sin(\theta/2)$, opposite the half angle. With θ in radians, the first two formulas become $s = r\theta$ and $\tfrac{1}{2}\theta r^2$.

A 60° slice of the 9-inch pan is 60 ÷ 360 = 1/6 of the cake. Its curved edge is 1/6 × 9π = 1.5π ≈ 4.7124 in long, its top is 1/6 × 20.25π = 3.375π ≈ 10.6029 in², and the straight cut across its outer edge is 4.5 in, the same as the radius, because a 60° slice makes an equilateral triangle.

Reading the result

• The headline is the number usually wanted from what you entered: the area from a radius or diameter, the diameter from a circumference, and the radius from an area.
• The three tiles give the other measurements. Your own measurement is marked “Given”; values with π in them also show the exact form, and in inches or feet the lengths come with the nearest 1/16 in.
• The note on measuring appears when you give a circumference and says how far a small tape-reading error moves the diameter.
• How this was calculated writes out each formula with your numbers, then the spreadsheet formulas that give the same values (for example =PI()*(9/2)^2 for the cake pan’s area).
• Your circle in other units lists all four values in inches, feet, millimeters, centimeters and meters (plus your unit), with the area in each unit’s square. It downloads as a CSV file.
• Half, double and triple reruns the calculation at those sizes. Doubling the pan’s diameter to 18 in gives 254.469 in² of area, four times as much, not twice.
• Save for comparison keeps up to three circles side by side. Save the 9-inch pan, change the diameter to 8 and save again, and the change column shows 13.3518 in² less area.
• Continue in the Length Converter carries the diameter over, to see it in inch fractions as fine as 1/64 in or in other length units.

Assumptions and limitations

• The shape is a perfect circle. Real pipes, posts and trunks are rarely perfectly round; an oval one measures a different diameter in each direction.
• Unless you choose 3.14 or 22/7, the decimals shown are rounded from the exact value with π to 100 decimal places, so every digit shown is right. The CSV files hold your browser’s numbers, which use π as 3.141592653589793 and agree to about 15 significant figures. Decimals are rounded for display only, no number shows more than 15 significant figures, and very large or very small values are written in scientific notation.
• All lengths use one unit and the area uses its square. The other units come from exact definitions such as 1 in = 2.54 cm.
• Measurements must be between 0.000000001 and 1,000,000,000,000 in the chosen unit; switch to a smaller or larger unit for anything outside that range.
• The central angle is in degrees and may be at most 360°. The calculator does not calculate segments (the part of a slice between the chord and the arc).

Common mistakes

• Using the diameter as the radius. A = πr² needs the radius. Putting the 9-inch pan’s diameter into the formula gives 81π ≈ 254.47 in², four times the real 63.62 in².
• Halving the circumference to get the radius. A 20-inch circumference means a radius of about 3.18 in, not 10 in. Divide by 2π, not by 2.
• Writing an area in plain units, or converting it with a length factor. Area is in square units: 63.62 in², not 63.62 in. And 1 in² is 2.54² = 6.4516 cm², so converting an area with 2.54 alone gives a figure 2.54 times too small.
• Expecting area to grow in step with the size. Twice the diameter is four times the area; the half, double and triple table shows it with your numbers.
• Mixing values of π. If your book uses 3.14, choose 3.14; otherwise a 7 cm radius gives 153.938 cm² where the answer key says 153.86 cm².
• Typing radians into a degrees box. The arc formula s = rθ uses radians, but the angle box takes degrees. A small decimal such as 1.0472 is probably radians; the calculator points this out and gives the degrees (60).