A circle with a radius of length \(r\) has an area that can be calculated using the formula

$$\large{A = \pi {r^2}}$$

Thus, we can easily calculate the area of a circle when the radius is given to us right away. There will be occasions in which the diameter is provided, rather than the radius. Because we know that the diameter is twice the radius, all we need to do to calculate the length of the radius is divide the diameter by \(2\).

Example 1: Find the area of the circle with a given radius.

Let's substitute the value of \(8\) for the radius in the formula and then simplify.

$$\eqalign{

A &= \pi {r^2} \cr

& = \pi {\left( 8 \right)^2} \cr

& = 64\pi \cr} $$

The **exact** area of the circle is \(64\pi\) m^{2}.

If we want an approximate value for the area, we can use the estimated numerical value of pi which is \(\pi = 3.1416\). This gives us

$$\eqalign{

A &= \pi {r^2} \cr

& = \left( {3.1416} \right){\left( 8 \right)^2} \cr

& = \left( {3.1416} \right)\left( {64} \right) \cr

& = 201.06 \cr} $$

The approximate area of the circle is about \(201.06\) m^{2.}

Example 2: Find the area of the circle with a given diameter.

The diameter of the circle is \(5\) cm. Since a diameter is twice the radius, we can divide it by \(2\) to get the length of the radius.

$$\eqalign{

r &= {1 \over 2}d \cr

& = {1 \over 2}\left( 5 \right) \cr

& = 2.5 \cr} $$

After we have determined the circle's radius (\(r = 2.5\)), determining the area of the circle is a relatively simple task.

$$\eqalign{

A& = \pi {r^2} \cr

& = \left( {3.1416} \right){\left( {2.5} \right)^2} \cr

& = \left( {3.1416} \right){\left( {2.5} \right)^2} \cr

& = \left( {3.1416} \right)\left( {6.25} \right) \cr

& = 19.64 \cr} $$

The area of the circle is about \(19.64\) cm^{2}.