# What is the area of a circle with circumference of 6.28?

Oct 1, 2015

Approximately $3.14$

#### Explanation:

The formula for circumference of a circle with radius $r$ is $2 \pi r$.

The formula for the area of a circle with radius $r$ is $\pi {r}^{2}$.

$\pi \approx 3.14$

So the radius of our circle is $\frac{6.28}{2 \pi} \approx \frac{6.28}{2 \cdot 3.14} = 1$

and its area is $\pi {r}^{2} \approx 3.14 \cdot {1}^{2} = 3.14$

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 $2 \pi r$.

To see that the area of a circle is $\pi {r}^{2}$ you can divide it 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...