Area Formulas

Counting every single square to find the area can take a long time. Let’s see if there is a shortcut.
Consider a rectangle that is 5 units long and 3 units wide.We can find its area by adding up the squares in each column.The area is \(\underbrace{\textcolor{colordef}{3}\;\;+\;\textcolor{colorprop}{3}\;\;+\;\textcolor{olive}{3}\;\;+\;\textcolor{purple}{3}\;\;+\;\textcolor{orange}{3}}_{5\,\text{times}} = 5 \times 3 = 15\) square units.
This shows that we can find the area of a rectangle by simply multiplying its length by its width.

This is a rectangle with length \(l = 8\) m and width \(w = 4\) m. Using the formula for the area of a rectangle:$$\begin{aligned}[t]A &= l \times w \\ &= 8 \times 4 \\ &= 32 \, \mathrm{m}^2\end{aligned}$$The area is \(32\) square meters (we read \(32\,\mathrm{m}^2\) as “32 square meters”).

To find the area of a triangle, we can cut it along its height to form two smaller triangles, then rearrange them to make a rectangle. Let’s see how this works step by step with the triangle below:

1. Cut the triangle along the height \(CH\) to form two smaller triangles. Rotate and rearrange these two triangles to form a rectangle:
2. The area of the rectangle is the length multiplied by the height: \(4 \times 3 = 12 \,\text{cm}^2\). Since the area of the rectangle is equal to twice the area of the original triangle, the area of the triangle is half the area of the rectangle:$$\begin{aligned}A_\triangle &= \dfrac{\text{base} \times \text{height}}{2}\\ &= \dfrac{4 \times 3}{2}\\ &= 6 \,\text{cm}^2 .\end{aligned}$$

$$\begin{aligned}[t]A &= \dfrac{b \times h}{2} \\ &= \dfrac{4 \times 2}{2} \\ &= 4 \, \text{cm}^2\end{aligned}$$So, the area of the triangle is \(4 \,\text{cm}^2\).

We can discover the formula for the area of a parallelogram by rearranging it into a rectangle.

1. Draw the height, which is a line from the top side to the bottom side that is perpendicular to the base:
2. Cut the triangle on the right side:
3. Move the triangle to the left side to form a rectangle:
4. Now we have a rectangle with a length (base) of \(4\) cm and a height of \(3\) cm. The area of the parallelogram is the same as the area of this rectangle, which is the base times the height: \(4 \times 3 = 12 \, \text{cm}^2\).

$$\begin{aligned}[t]A &= b \times h \\ &= 4 \times 2 \\ &= 8 \, \text{cm}^2\end{aligned}$$So, the area of the parallelogram is \(8 \,\text{cm}^2\).

To find the area of a circle, we can divide it into smaller parts and rearrange them to approximate a parallelogram. Let’s see how this works step by step:

1. Divide the circle into 12 equal parts, like slices of a pie:
2. Imagine cutting these 12 parts from the circle.
3. Rearrange the parts by alternating them to form a shape that looks like a parallelogram:
4. The base of the parallelogram is approximately half the circumference of the circle (\(\pi r\)), and its height is approximately the radius (\(r\)). So, the area of the circle is the area of the parallelogram:$$\begin{aligned}A_\text{circle} &= (\pi r) \times r \\ &= \pi \times r \times r \\ &= \pi r^2.\end{aligned}$$We read this as “pi \(r\) squared”.

$$\begin{aligned}[t]A &= \pi \times r \times r \\ &= \pi \times 3 \times 3 \\ &= 9\pi \\ &\approx 28.3 \, \text{cm}^2 \quad (\text{using } \pi \approx 3.14)\end{aligned}$$

$$\begin{aligned}[t]A &= \textcolor{colordef}{\text{Area of square}} + \textcolor{colorprop}{\text{Area of triangle}} \\ &= \textcolor{colordef}{s \times s} + \textcolor{colorprop}{\dfrac{b \times h}{2}} \\ &= \textcolor{colordef}{3 \times 3} + \textcolor{colorprop}{\dfrac{3 \times 2}{2} }\\ &= \textcolor{colordef}{9} + \textcolor{colorprop}{3} \\ &= 12 \, \text{cm}^2\end{aligned}$$