Question

What is the circumference of a unit circle?

Answer 1

`2pi`

Explanation:

The radius of a unit circle is `1` and its diameter is `2`.

`pi` can be defined as the ratio of the circumference of a circle to its diameter, hence the circumference of the unit circle is `2pi`.

In general the circumference of a circle of radius `r` is `2pi r`.

The area of the unit circle is `pi` (in general `pi r^2` where `r` is the radius).

My favourite way of deriving that is to imagine the circle as a round cake, which you cut radially into a large number of equal sized slices. You then reassemble these pieces into a (nearly) rectangular shape by alternating them, side by side, head to tail. This rectangular arrangement will have longer side equal to half the circumference and shorter side equal to the radius, hence total area:

`r * (2pi r)/2 = pi r^2`