Area and perimeter of a circumference

Area

The curve called circumference contains a surface. This surface is called the area of the circumference.

There is a very simple formula that allows us to calculate the area enclosed within the circumference just by knowing the length of the radius of the circle.

Let us call $r$ the radius of the circle, then the area of the circle is:

$$A=\pi\cdot r^2$$

Remember that $\pi$ is an irrational number, so if we want to express the result of the area without the constant $\pi$, we have to calculate it using the approximation $\pi=3,1416$

Let's see an example of how we calculate the area of a circle.

In the circumference of the image shown above, it is clear that the area enclosed by the circle is the blue area. In this case the variable $r$, the length of the radius, takes the value $r = 10$cm. The area is calculated as follows:

$$A=\pi\cdot r^2= \pi \cdot 10^2= 3,1416 \cdot 100=314,16 \mbox{ cm}^2$$

Note 1: we can see that the units of the parameter $r$ are cm. It could be any unit of measurement, such as cm, m, mm... or other units such as inches or miles, for example.

Note 2: the area units are units of squared length because we have multiplied a distance by itself.

Perimeter

Consider a circumference, the perimeter of a circle is the length of the curve; in other words, it is the distance that a person would walk if he started walking from any point on the circumference and gave a whole lap around the circumference until arriving at the point of departure.

In a similar way as with the area, there is an expression that allows us to know the length (or perimeter) of the circumference simply by knowing its radius $r$.

The expression is:

$$P=2\cdot \pi \cdot r$$

Let's see it more clearly with an example.

Take the circumference of the example above, and we represent it again:

Again the parameter $r$, the length of the radius is $r = 10$ cm.

$$P=2\cdot \pi \cdot r= 2 \cdot \pi \cdot 10= 2 \cdot 3,1416 \cdot 10 = 62,832 \mbox{ cm}$$

Therefore, the result is that the perimeter is $62,832$ cm.