Circumference Calculator

Calculate the circumference of a circle. C = 2πr = πd. Convert between circumference, radius, and diameter with step-by-step solutions.

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Result

Circumference
31.415927
cm
Radius5 cm
Diameter10 cm
Circumference31.415927 cm
Area78.539816 cm²

Step-by-Step Solution

C = 2πr = 2π(5) = 31.415927 cm

What is Circumference?

Circumference is the distance around the outside of a circle. It is the circle's equivalent of a polygon's perimeter. The word comes from the Latin "circumferre" meaning "to carry around." The circumference is directly proportional to the circle's radius through the mathematical constant π (pi), which is approximately 3.14159265358979.

Every circle, regardless of size, has the same ratio of circumference to diameter. That ratio is exactly π. This remarkable fact was known to ancient civilizations and remains one of the most fundamental constants in all of mathematics.

Circumference Formulas

From Radius

Multiply two, pi, and the radius together.

C = 2πr

From Diameter

Multiply pi by the diameter directly.

C = πd

Radius from Circumference

Divide the circumference by 2π.

r = C / (2π)

Diameter from Circumference

Divide the circumference by π.

d = C / π

From Area

Derive circumference from the enclosed area.

C = 2√(πA)

Area from Circumference

Derive the area from the circumference.

A = C² / (4π)

Step-by-Step: How to Calculate Circumference

Method 1: Using the Radius

  1. Identify the radius of the circle (the distance from center to edge).
  2. Multiply the radius by 2 to get the coefficient.
  3. Multiply the result by π (3.14159...).
  4. The answer is the circumference in the same unit as the radius.

Method 2: Using the Diameter

  1. Identify the diameter (the distance across the circle through the center).
  2. Multiply the diameter by π (3.14159...).
  3. The answer is the circumference in the same unit as the diameter.

Method 3: Reverse Calculation (Finding Radius/Diameter from Circumference)

  1. If you know the circumference and need the radius: divide by 2π.
  2. If you need the diameter: divide the circumference by π.
  3. To find the area from the circumference: A = C² / (4π).

Circumference in the Real World

  • Tire Size: Tire circumference determines how far a vehicle travels per revolution. A tire with 70 cm diameter has C = π(70) = 219.91 cm per revolution.
  • Earth's Circumference: The Earth's equatorial circumference is approximately 40,075 km. This was first estimated by Eratosthenes around 240 BC using shadow angles.
  • Pipework: Pipe circumference is needed to calculate insulation wrap, gasket sizes, and heat transfer rates.
  • Sports: A standard basketball has a circumference of about 75 cm. A soccer ball is about 69 cm. Track curves are arc lengths.
  • Cooking: Cake pan circumference helps determine decoration length. A 9-inch pan has C = 9π = 28.27 inches of edge.

Common Circumference Values

Here are some commonly encountered circumference calculations for quick reference:

  • r = 1: C = 6.2832
  • r = 2: C = 12.5664
  • r = 5: C = 31.4159
  • r = 10: C = 62.8318
  • r = 25: C = 157.0796
  • r = 100: C = 628.3185

Circumference and Pi

The number π is defined as the ratio of a circle's circumference to its diameter: π = C/d. It is an irrational number, meaning its decimal expansion never terminates or repeats. The first 20 digits are 3.14159265358979323846. For most practical applications, using π = 3.14159 provides more than sufficient accuracy. For engineering calculations, 3.14159265 (8 decimal places) is commonly used.