Pi Calculator

A circle with radius 5 cm has a circumference of 31.4 cm and an area of 78.5 cm²

With pi taken as 3.14, a circle of radius 5 cm has circumference 10 cm × 3.14 = 31.4 cm and area 5 × 5 × 3.14 = 78.5 cm². Keep pi as π and the circumference is about 31.415927 cm and the area about 78.539816 cm², slightly larger, because 3.14 is about 0.0016 below π (3.14159…).

The calculator above takes any one of radius, diameter, circumference or area and fills in the other three. Under "Value of pi" you can pick π, 3.14, 3.1, 3 or 22/7, whichever your problem specifies; the results and the working change to match, and you can compare all five side by side. Values for 3.14 and 22/7 are computed as exact fractions, so an area of 78.5 gives a radius of exactly 5, not 4.999….

How to use it

1. Enter what you know: type one of radius, diameter, circumference or area into its field. That field is marked "entered" and the other three become calculated values. Type in a different field and that one becomes the new starting point.
2. Pick a value of pi: if the problem says "use 3.14", choose 3.14. Every row in the list already shows the result for its value of pi, so you can see at a glance how far 3.14 is from π. The up and down arrow keys also move the selection.
3. Unit: choose cm, m, in or similar and it is appended to the results, with a square unit such as cm² for area. Changing the unit does not convert the numbers.
4. Figures: "Unroll circumference" rolls the circle through one turn to show the circumference is about 3.14 diameters. "Unfold area" cuts the circle into slices and lays them alternately into a shape that approaches a rectangle; try 8 to 64 slices, and use "Replay" to watch again.
5. Working: the working box shows the equation with your numbers, such as "Circumference = diameter × π = 10 × 3.14 = 31.4 cm". If you entered a radius or diameter, answers left in terms of π, such as 10π cm and 25π cm², appear as well.
6. How is pi found?: at the bottom you can add sides to a regular polygon to squeeze π, or measure a cup's circumference and diameter and see how close circumference ÷ diameter gets. "Copy result" copies the four values; "Copy link" copies an address containing your input.

How it is calculated

What pi means

Pi is a circle's circumference divided by its diameter. For a coin or a wheel, circumference ÷ diameter is always the same number, 3.14159265…, written with the Greek letter π. Roll a circle through one turn and it covers 3 diameters plus about 0.14 of another, which is exactly what the "Unroll circumference" figure shows.

Circumference

Since pi = circumference ÷ diameter, circumference = diameter × pi, or C = πd. The diameter is twice the radius, so C = 2πr. For a radius of 5 cm that is 5 × 2 × 3.14 = 31.4 cm.

Area

Area = radius × radius × pi, or A = πr². Cut the circle into thin slices and lay them alternately up and down: the shape approaches a rectangle as wide as half the circumference (πr) and as tall as the radius (r). More slices make it straighter, so the area is width × height = πr × r = πr². For a radius of 5 cm that is 5 × 5 × 3.14 = 78.5 cm².

Working backwards

From the circumference, diameter = circumference ÷ pi, and half of that is the radius. From the area, area ÷ pi = radius × radius, and the radius is the square root of that. For example, an area of 78.5 cm² with pi = 3.14 gives 78.5 ÷ 3.14 = 25, and since 5 × 5 = 25 the radius is 5 cm. The same area worked back with π gives a radius of about 4.998732 cm, so to get a clean answer, work backwards with the same value of pi the problem used.

Circumference and area by radius

The π columns are rounded to six decimal places. For radius 2 cm the circumference and area show the same number, and that is no accident: 2πr and πr² are equal when r = 2, although the units differ (cm and cm²).

Which value of pi should you use?

If a problem or an assignment tells you which value of pi to use, use that one. Without instructions, 3.14 is the usual everyday choice. If you are asked to leave the answer in terms of π, such as 25π cm², see the π form under the working. The table shows how much the choice matters.

• 3.14: the common choice for school work and quick estimates. Results come out about 0.05% below those computed with π.
• 3 or 3.1: for mental estimates. Using 3 puts you about 4.5% short, so a string cut to a circumference computed with it will not reach around.
• 22/7: when the radius or diameter is a multiple of 7, the 7 cancels and the arithmetic is tidy. For diameter 14 cm the circumference is 14 × 22/7 = 44 cm and the area is 7 × 7 × 22/7 = 154 cm². It is about 0.04% above π.
• π as is: for cutting material to fit or setting design dimensions, where accuracy matters, calculate with π. This calculator uses the browser's Math.PI (3.141592653589793).

How was pi found?

The simplest way is to measure. Wrap string once around a cup or can, measure it, measure the diameter with a ruler and divide, and you get something near 3.1. But a 1 mm slip on a cup 8 cm across changes the result by about 1.25%, so measuring alone cannot pin down the second decimal place.

In the 3rd century BC, Archimedes closed in on π by calculation instead. Draw a regular hexagon inside a circle and another outside it, and the circle's circumference lies between their perimeters. Keep doubling the number of sides and both perimeters close in on the circle; at 96 sides he proved 223/71 < π < 22/7, that is 3.1408 < π < 3.1429. The table below lists the perimeters for a circle of diameter 1; the "How is pi found?" section under the calculator takes it all the way to 24576 sides.

Each time the number of sides doubles, the gap shrinks to about a quarter. Archimedes' 223/71 and 22/7 are a little wider than the 96-gon values in the table because, along the way, he approximated values such as √3 by fractions and rounded in the direction that widens the range. Around AD 265, Liu Hui of Wei in China took the same idea to 3072 sides and got π ≈ 3.1416. Around AD 480, Zu Chongzhi found 3.1415926 < π < 3.1415927 and left behind the fraction 355/113, which agrees with π to six decimal places.

That π never ends was established in 1768, when Lambert proved it irrational, and in 1882 Lindemann proved it transcendental, meaning it is not a root of any polynomial with integer coefficients. It follows that a square with exactly the same area as a given circle cannot be drawn with compass and straightedge alone. Today, computers using rapidly converging series have calculated π to hundreds of trillions of digits.

The first 100 digits of pi

Here are the first 100 decimal places of pi in groups of ten. Everyday work needs only 3.14 or 3.1416, and the "π as is" setting here uses about 15 decimal places of precision.

3.1415926535 8979323846 2643383279 5028841971 6939937510
  5820974944 5923078164 0628620899 8628034825 3421170679

Common mistakes

• Using the diameter for area: for a circle with diameter 10 cm, 10 × 10 × 3.14 = 314 is four times too big. Use the radius: 5 × 5 × 3.14 = 78.5 cm². If a problem gives the diameter, halve it first.
• Mixing up the two formulas: circumference is diameter × pi (radius × 2 × pi); area is radius × radius × pi. Radius × pi alone gives half the circumference.
• Forgetting that area is squared: if lengths are in cm, area is in cm². Converting to m, 1 m² = 10,000 cm², so dividing by 100 as you would for a length is wrong.
• Ignoring the value of pi the problem specifies: if the question says "use 3" and you use 3.14, the area of a 5 cm circle comes out as 78.5 cm² instead of 75 cm², and the answer is wrong.
• Rounding in the middle: from a circumference of 20 cm, rounding the diameter 6.369… to 6.4 before finding the area gives 32.1536 cm², about 0.3 cm² above the unrounded 31.847134 cm² (pi = 3.14). Round once, at the end.

Limits

• The unit is only a label on the results and is never converted. If you enter a radius in cm and need metres, use a unit converter as well. All four fields are in the same unit, and area is in that unit squared.
• Radius, diameter and circumference can range from 0.001 to 1,000,000, area up to 1,000,000,000, with up to six decimal places, and every value must be greater than 0.
• Results are rounded to six decimal places for display, and values below 1 show six significant digits. Values for 3.14, 3.1, 3 and 22/7 are computed as exact fractions; where a decimal does not terminate or has been rounded, ≈ is shown.
• "Measure and divide" simply divides what you enter and does not correct for measurement error. The figures for pi and the polygons are illustrations and are not drawn to scale for your input.
• Circular arcs and sectors, the surface area and volume of cylinders and spheres, and ellipses are not covered. Your input is calculated in your browser only and is not sent anywhere; "Copy link" does put your input in the address.

Basis: the value of pi and the circle formulas do not change. The sources below were checked against the originals on October 21, 2026.

Sources

• 01Wikipedia, Piπ as circumference ÷ diameter; Archimedes' 96-gon 223/71 < π < 22/7; Liu Hui's 3072-gon 3.1416; Zu Chongzhi's 3.1415926 < π < 3.1415927 and 355/113; Lambert (1768) irrational and Lindemann (1882) transcendental
• 02Wolfram MathWorld, Archimedes' AlgorithmOuter perimeter 2n tan(π/n), inner 2n sin(π/n); starting from the hexagon and doubling with harmonic and geometric means; 96-gon 3.14103 < π < 3.14271
• 03Wolfram MathWorld, Pi Approximations3, 22/7, 333/106, 355/113 … are correct to 0, 2, 4 and 6 decimal places respectively
• 04OEIS A000796, Decimal expansion of PiDecimal expansion of pi (checked against the 100 digits on this page)

Frequently asked questions

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