Circumference Calculator
Calculate the circumference and area of a circle using its radius or diameter.
Results
Circle Formulas
Circumference: C = 2πr = πd
Area: A = πr²
Diameter: d = 2r
Where r = radius, d = diameter, and π ≈ 3.14159...
What Is Pi (π)?
Pi (π) is the ratio of a circle's circumference to its diameter. It's a mathematical constant approximately equal to 3.14159. Pi is an irrational number, meaning its decimal representation never ends and never repeats.
π ≈ 3.14159265358979...
Common Circle Measurements
| Radius | Diameter | Circumference | Area |
|---|---|---|---|
| 1 | 2 | 6.28 | 3.14 |
| 2 | 4 | 12.57 | 12.57 |
| 5 | 10 | 31.42 | 78.54 |
| 10 | 20 | 62.83 | 314.16 |
Arcs, Sectors, and Segments
A full circle has a circumference of 2πr. But circles can also be divided into arcs and sectors, each with their own formulas derived from the full circle.
Arc Length
The length of a portion of the circumference determined by a central angle θ:
Arc Length = (θ / 360°) × 2πr [degrees]
Arc Length = θ × r [radians]
Sector Area
The area of a "pie slice" defined by central angle θ:
Sector Area = (θ / 360°) × πr² [degrees]
Sector Area = ½ × θ × r² [radians]
Example: 90° sector of radius 6
Arc = (90/360) × 2π × 6 = 0.25 × 37.70 ≈ 9.42
Area = (90/360) × π × 36 = 0.25 × 113.10 ≈ 28.27
Real-World Applications of Circumference
🚗 Wheels & Tires
A car tire with diameter 70 cm travels C = π × 70 ≈ 219.9 cm per revolution. Odometers count wheel revolutions to calculate distance traveled.
🌍 Earth & Astronomy
Earth's circumference ≈ 40,075 km (equatorial). Eratosthenes calculated this in 240 BC using shadow angles and distance between cities — accurate to within 2%.
🏃 Running Tracks
Standard 400 m track designs use circumference formulas for the curved sections. Each lane is wider, so outer lane runners start at a stagger offset = 2π × (lane width).
🔧 Engineering & Manufacturing
Pipe circumference determines the amount of material needed for wrapping or insulation. Belt-and-pulley systems use circumference ratios to calculate gear ratios and speed.
History of Pi
The quest to understand π spans over 4,000 years of mathematical history. Ancient civilizations approximated the ratio long before it was proven to be irrational.