\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Area

What is Area?

Definition Area
The area of a shape is how much space it covers on a flat surface.
We measure area by counting how many square units fit inside the shape.
To find the area of a shape, we can place it on a grid and count the total number of squares it covers.
You can think of it like tiling a floor — the area is the total number of tiles you use.
Example
Find the area of the green shape. Each small square in the grid is 1 square unit.

To find the area, we count each square unit inside the shape.
There are 4 small squares inside the shape.
So, the area is 4 square units.

Units of Area


When we measure area, it is important to use standard units so that everyone gets the same measurement. Non-standard units, such as books or tiles of different sizes, can give different answers because they are not all the same size. For area, we use standard units like the square centimeter and the square meter.

Definition Units of Area
  • Square centimeter \(\left(\mathrm{cm}^2\right)\): The area of a square that is 1 cm on each side. This is about the size of your thumbnail. We write this as \(1\;\text{cm}^2\).
  • Square meter \(\left(\mathrm{m}^2\right)\): The area of a square that is 1 m on each side. This is about the size of the top of a small square table. We write this as \(1\;\text{m}^2\).

Area of Rectangles and Squares


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.

Proposition Area Formulas
To find the area of a rectangle, multiply its length by its width.To find the area of a square, multiply the side length by itself.
Shape Diagram Area Formula
Rectangle \(A = l \times w\)
Square \(A = s \times s\)
Example
Find the area of the rectangle:

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”).