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

Approximately ##3.14##

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 ~~ 3.14##

So the radius of our circle is ##6.28 / (2 pi) ~~ 6.28 / (2 * 3.14) = 1##

and its area is ##pi r^2 ~~ 3.14 * 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...