Circumference Calculator
Enter the radius or diameter of a circle to calculate its circumference and area using the formulas C = 2πr and A = πr².
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Formula
C = 2πr | A = πr²
The circumference C equals 2 times π (approximately 3.14159) times the radius r. Area A equals π times radius squared. If you know the diameter instead, circumference = π × diameter. These formulas have been known since ancient times — Archimedes proved that π is between 3 10/71 and 3 1/7.
How to use the Circumference Calculator
- 1
Enter your radius
Value should be in units.
- 2
Read your results instantly
Results update in real time as you type.
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Pi and why it appears everywhere
Pi (π) is the ratio of any circle's circumference to its diameter. No matter the size — from atomic to galactic — this ratio is always exactly π ≈ 3.14159265358979... It is irrational (non-terminating, non-repeating) and transcendental (not the root of any polynomial with rational coefficients).
Pi appears far beyond circles. It shows up in probability (Buffon's needle problem), statistics (the normal distribution), quantum mechanics, and number theory. Euler's identity, e^(iπ) + 1 = 0, is often called the most beautiful equation in mathematics.
Circumference vs. perimeter
Circumference specifically refers to the perimeter of a circle. The concept generalizes: an ellipse has a circumference too, though the formula is much more complex. For a circle, the relationship between circumference and area gives us: A = C² / (4π) — area can be derived from circumference alone.
In everyday life, circumference matters for anything round: tire size (measured by circumference), track and field (a standard 400m running track has an inner circumference of exactly 400m), wheels, and pipes.
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Radians and the unit circle
Radians are the 'natural' unit for angles when working with circles. One radian is the angle subtended by an arc length equal to the radius. Since circumference = 2πr, there are 2π radians in a full circle (360°).
To convert: degrees × π/180 = radians. Radians → degrees: × 180/π. This system simplifies calculus, Fourier transforms, and most advanced math by eliminating the arbitrary factor of 360.
Tips & Insights
Circumference from diameter is simpler
If you measure a circular object's diameter (easiest to measure directly), just multiply by π: C = πd ≈ 3.14159 × d. No need to halve first.
Area grows with the square of radius
Doubling the radius quadruples the area. A pizza with a 16" diameter has 4× the area of an 8" pizza — not just 2×. This is why larger sizes are disproportionately better value.
Worked Examples
Bicycle wheel
Circumference: 213.6 cm (2.136 m). This means every wheel revolution moves the bike exactly 2.136 meters forward — useful for calibrating bike computers.
Round table
Circumference: 471.2 cm (about 15.5 feet). Area: 17,671 cm² (about 19 square feet). For a tablecloth with 30cm overhang, you'd need a 210cm diameter cloth.
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Frequently Asked Questions
What is circumference?
Circumference is the distance around the outside of a circle. It equals 2 × π × radius, or approximately 6.283 × radius.
What is the value of pi?
π ≈ 3.14159265358979... It is irrational and transcendental — its decimal representation never terminates or repeats. For most calculations, 3.14159 provides sufficient accuracy.
How do I find the radius from the circumference?
r = C ÷ (2π). For example, if circumference is 31.416 cm: r = 31.416 ÷ 6.2832 = 5 cm.
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