Area of Circle Formula
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. |
The area of a circle is given by multiplied 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.
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.
Set up the formula.
Subsitute the radius in for the variable .
Evaluate the exponent expression.
Evaluate the multiplication expression.
The area of the circle is equal to units squared.
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.
Set up the formula.
Subsitute the radius in for the variable .
Evaluate the exponent expression.
Evaluate the multiplication expression.
The area of the circle is equal to units squared or approximately units squared.
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.
Set up the formula.
Subsitute the radius in for the variable .
Evaluate the exponent expression.
Evaluate the multiplication expression.
The area of the circle is equal to units squared or approximately units squared.
The area of circle formula can be derived multiple ways using calculus[1] or using a visual proof as shown below.
The area of circle formula can be visually proven by dividing the area into concentric 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 illustrations below:
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.
The circle constant τ (tau) is a geometric constant approximately equal to 6.283. The numeric value is defined as the length of any circle's circumference divided by the length of its radius.
The area of a triangle is given by one-half multiplied by its width and height.
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Derive Area of Circle FormulaWumbo (internal)