Question

The area of a circle is 16pi. What is the circumference of the circle?

Answer 1

`8pi`

Explanation:

The area of a circle is `pir^2` where `r` is the radius.

So we are given:

`pir^2 = 16pi`

Dividing both sides by `pi` we find `r^2=16=4^2` and hence `r=4`.

Then the circumference of a circle is `2pir` so in our case:

`2pir = 2*pi*4 = 8pi`

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Footnote

Why is the circumference and area of a circle given by these formulas?

First note that all circles are similar and hence the ratio of the circumference to the diameter is always the same. We call that ratio, which is approximately `3.14159265`, `pi`. Since the diameter is twice the radius, we get the formula `2pir`.

To see that the area of a circle is `pi r^2` 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...