Area of Circle Formula

This figure illustrates the area of circle formula.

Formula

Summary

The area of a circle is given by mutiplied by the circle constant (tau) multiplied by the radius of the circle squared.

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

The area of the circle with a radius of length is equal to units squared. The value can be substituted into the expression to calculate an area of approximately units squared.

Examples

Area of Circle Radius 1

area of circle radius 1

To calculate the area of a circle given a radius of one, set up the equation for the area of the circle and substitute the value of the radius into the equation.

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

  2. Evaluate the exponent expression.

  3. Evaluate the multiplication expression.

  4. The area of the circle is equal to units squared.

Area of Circle Radius 2

area of circle radius 2

To calculate the area of a circle given a radius of one, set up the equation for the area of the circle and substitute the value of the radius into the equation.

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

  2. Evaluate the exponent expression.

  3. Evaluate the multiplication expression.

  4. The area of the circle is equal to units squared or approximately units squared.

Area of Circle Radius 3

area of circle radius 3

To calculate the area of a circle given a radius of one, set up the equation for the area of the circle and substitute the value of the radius into the equation.

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

  2. Evaluate the exponent expression.

  3. Evaluate the multiplication expression.

  4. The area of the circle is equal to units squared or approximately units squared.

Explanation

The area of circle formula can be derived multiple ways such as using calculus[1] or using a visual proof as shown below.

Visual Proof

The area of circle formula can be visually proven by dividing the area into cocentric rings. Unwrapping the rings and placing them next together forms the shape of a triangle. As the number of rings approaches infinity their area approaches the area of the triangle that they form. This is shown in the series of illustration below:

area of circle visual proof n=3 area of circle visual proof n=5 area of circle visual proof n=10 area of circle visual proof n=infinity

Recall that the area of a triangle is given by the formula:

Then, we can substitute the length of the base and height into the formula. The base is equal to the circumference of the circle which is given by the circle constant multiplied by the radius . The height is equal to the radius of the circle. This visual proof gives us the area of the circle formula.

Links

References

  1. Derive Area of Circle Formula
    Wumbo (internal)