# Why is the area of a circle given by the formula pir^2 ?

Aug 31, 2016

See explanation...

#### Explanation:

The circumference of a circle is $\pi$ times its diameter. In fact that's the original definition of $\pi$. SInce the diameter is twice the radius, that means that the circumference of a circle of radius $r$ is $2 \pi r$.

If we take a circle of radius $r$ and cut it into a number of segments, we can then reassemble those segments into a sort of bumpy parallelogram with height $r$ and longer sides of length $\pi r$ (each being half the circumference).

If we use a larger number of thin segments, then this is more like a rectangle with height $r$ and base $\pi r$, which therefore has area $r \cdot \pi r = \pi {r}^{2}$

Here's an animation for just $8$ segments...

So the area of the circle is proportional to the square of the radius.