Circumference Calculator

What Is Circumference?

The circumference is the distance around the outside of a circle, its perimeter.
Cut the edge of a circle and roll it out flat, and the circumference is how long that line turns out to be.

The Formulas

From the radius (r): C = 2πr

From the diameter (d): C = πd

To find radius from circumference: r = C / (2π)

To find diameter from circumference: d = C / π

Area of a circle: A = πr²

The Number Pi (π)

π (pi) is the ratio of any circle’s circumference to its diameter, and it is the same number for every circle no matter how big.
π is an irrational number, meaning its decimal expansion never ends and never falls into a repeating pattern:

π ≈ 3.14159265358979…

The history of π spans thousands of years:

• Archimedes of Syracuse (287–212 BC) was the first to rigorously calculate π, using inscribed and circumscribed polygons, determining it lies between 3 10/71 and 3 1/7 (approximately 3.1408 to 3.1429)
• William Jones first used the Greek letter π for this ratio in 1706
• Leonhard Euler popularized the notation in 1737

Today, π has been calculated to over 100 trillion decimal places, though only 39 digits are needed to calculate the circumference of the observable universe to the accuracy of a hydrogen atom.

Radius vs. Diameter

• Radius: distance from the center to the edge
• Diameter: distance across the full circle through the center
• Always: d = 2r

Practical Applications

• Tire circumference: A car tire with a 14-inch radius has C = 2π × 14 ≈ 87.96 inches per revolution. This is how your odometer calculates distance.
• Circular garden: a bed with a 3-meter radius has a circumference of about 18.85 meters, which is how much edging you need to buy.
• Pipes and tubes: the circumference of a pipe tells you how much lagging or wrap it takes to cover.
• Wheel rotations: distance ÷ circumference = how many times the wheel turns. A 2-meter circumference wheel covers 2 meters per turn, so 100 meters takes 50 rotations. Getting this the wrong way up is the classic slip, and it gives you an answer that shrinks as the wheel gets bigger.

Unit Conversion Reference

Worked Example

A circle has a radius of 5 inches:

• Circumference = 2 × π × 5 = 31.416 inches
• Diameter = 2 × 5 = 10 inches
• Area = π × 5² = 78.540 square inches