### The formula for area of a circle

The dissections shown above strongly suggest that the correct formula for the area of a circle with radius r is

A formal proof of this result requires an understanding of the concept of limits, which will be studied in senior mathematics.

#### Example 1

- Find the area of a circle whose radius is 7 cm.
- Find the area of a circle whose diameter is 7 cm.

Give each answer:

- in terms of π
- as an approximate value, using \(\pi \approx \dfrac{22}{7}.\)

#### Solution

\begin{align}{\bf \text{a i}}\hspace{22px} A &= \pi r^2\\\\ &= 49\pi\ \text{cm}^2\end{align} \begin{align}{\bf \text{a ii}}\hspace{18px} A &= 49\pi\\\\ &\approx 49 × \dfrac{22}{7}\\\\ &= 154\ \text{cm}^2\end{align}b i Since the diameter is 7 cm, the radius is \(\dfrac{7}{2}\ \text{cm}\)

\begin{align}A &= \pi r^2\\\\
&= \dfrac{49}{4}\pi\ \text{cm}^2\end{align}

\begin{align}{\bf \text{b ii}}\hspace{18px} A &=\dfrac{49}{4}\pi\\\\
&\approx \dfrac{49}{4} × \dfrac{22}{7}\\\\
&=38\dfrac{1}{2}\ \text{cm}^2\end{align}
#### Example 2

A circle has area \(1386\ \text{cm}^2.\) Using \(\pi \approx\dfrac{22}{7},\) find the approximate value of the radius.

#### Solution

\begin{align}A &= \pi r^2\\\\ \pi r^2 &= 1386\\\\ r^2 &= \dfrac{1386}{\pi}\\\\ &\approx 1386 ÷\dfrac{22}{7} \\\\ &\approx \dfrac{1386}{1}× \dfrac{7}{22} \\\\ &= 441\\\\ r &= \sqrt{441}\\\\ &= 21\ \text{cm}\end{align}#### Example 3

Four semicircles are drawn along the edge of a square with side length 14 cm.

- Find the area of the entire region, giving your answer in terms of π.
- Find the approximate value of the area, using \(\pi \approx \dfrac{22}{7}\) .

#### Solution

- Area region = area of the square + 2 × area of the circle

\begin{align}\text{Area square} &= 14 × 14\\\\ &= 196\ \text{cm}^2\end{align} \begin{align}\text{Radius of circle} &= 7\ \text{cm}\end{align} \begin{align}\text{Area of circles} &= 2 × \pi × 7^2\\\\ &= 2 × \pi × 49\\\\ &= 98\pi\ \text{cm}^2\end{align} Hence the area of the region = (196 + 98π) cm\(^2\) - \begin{align}\text{Area} &≈ 196 + 98 × \dfrac{22}{7}\\\\ &= 196 + 308\\\\ &= 504\ \text{cm}^2\end{align}