Circumference of Circle Formula

This figure illustrates the circumference of a circle formula.

Formula

Summary

The circumference of a circle is given by the circle constant (tau) multiplied by the radius.

Variable Description
Circumference of the circle
The circle constant (tau), where
The radius of the circle
Note: This website uses the constant (tau) instead of (pi) as the default circle constant. The substitution can be used to translate between the two constants.

Usage

The circumference of a circle is given by the constant (tau) multiplied by the radius of the circle. For example, to find the circumference of a circle with a radius of length the formula is:

To find a numeric answer, substitude the value in for the circle constant and evaluate the multiplication.

The circumference of a circle with a radius of length is equal to units.

Examples

Example 1

circle radius 1

This example demonstrates how to find the circumference of a circle with a radius of using the circumference formula.

Steps

  1. Set up the formula and substitute the radius in for the variable .

  2. Substitute in for the circle constant.

  3. Evaluate the multiplication.

  4. The circumference of the circle with a radius of is equal to units or approximately units.

Example 2

circle radius 2

This example demonstrates how to find the circumference of a circle with a radius of using the circumference formula.

Steps

  1. Set up the formula and substitute the radius in for the variable .

  2. Substitute in for the circle constant.

  3. Evaluate the multiplication.

  4. The circumference of the circle with a radius of is equal to units or approximately units.

Example 3

circle radius 3

This example demonstrates how to find the circumference of a circle with a radius of using the circumference formula.

Steps

  1. Set up the formula and substitute the radius in for the variable .

  2. Substitute in for the circle constant.

  3. Evaluate the multiplication.

  4. The circumference of the circle with a radius of is equal to units or approximately units.

Explanation

The formula for the circumference of a circle in terms of its radius comes from the definition of the circle constant (tau). The circle constant is defined as the ratio of any circles circumference divided by its radius. This is illustrated in the figure below.

Definition of the circle constant τ (tau)
Figure 1: τ (tau) Definition

Derivation

To derive the formula, start with the definition of the circle constant.

Multiply both sides of the equation by .

The result is the circumference formula.

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