# What is the formula for the area of a circle?

Mar 3, 2017

$A = \pi \cdot {r}^{2} =$ area of a circle

Mar 3, 2017

The formula $A = \pi {r}^{2}$ expresses the area of a circle in terms of its radius.

#### Explanation:

The number $\pi$ is defined as the ratio of the circumference of a circle to its diameter (i.e. to twice its radius), hence the formula $2 \pi r$.

The area of a circle is given by the formula:

$A = \pi {r}^{2}$

To see why, you can divide a circle into a number of equal segments and stack them head to tail to form a sort of parallelogram with 'bumpy' sides. The long sides will be about half the circumference in length - that is $\pi r$, while the height of the parallelogram will be about $r$. So the area is seen to be about $\pi {r}^{2}$.

This approximation gets better the more segments you have, but here's an animated illustration I put together with just $8$ segments...