Introduction
Below, a rectangle and a square are drawn with markings on their edges. Edges with the same markings show that they are the same size.
Area of rectangles and squares
The area of a rectangle is the amount of space that can fit within its outline. One method of finding the area is to divide a rectangle into unit squares and to count the number of these squares.
A unit square is defined to be a square with a side length of 1 unit, and a single unit square has an area of 1 \text{ unit}^2. In this way, counting the number of unit squares in a shape tells us the area of that shape as a multiple of 1 \text{ unit}^2.
Let's look at an example.
Similarly, this method can be used to find the area of a square.
You may have noticed that multiplying the dimensions of a rectangle gives its area. Are there different pairs of numbers that multiply to give the same answer? In other words, is it possible for rectangles to have the same area when they have different dimensions?
For example, suppose we knew only that a rectangle had an area of 4 \text{ mm}^2. What rectangles could we create using unit squares that have this area?
We can start with 4 identical unit squares, each with side length 1 mm, and find the ways we can arrange them so that they form a rectangle. Here are the three rectangles we can form that have whole number side lengths and an area of 4 \text{ mm}^2.
We can draw three rectangles because there are three pairs of whole numbers that multiply to give an answer of 4 \text{ mm}^2. Notice that 1 \times 4 is the same as 4 \times 1, so the first two rectangles are the same, but one is a rotated version of the other.
Exploration
Use the applet below to draw rectangles with a target area. The left side of the applet will tell you how many rectangles can be drawn. Once you have drawn the required number of rectangles, click Next Game.
As an added challenge, you can click Hide Grid to draw rectangles by thinking of two numbers that multiply to give the target area, rather than counting the unit squares.
We can draw different rectangles with different dimensions but with the same area. As we increase the number of unit squares, the area of the rectangle also increases.
While dividing rectangles into unit squares is effective for finding the area, it can be time consuming, especially for larger rectangles. The method of multiplying the dimensions of a rectangle is the quickest way to find its area. The formula for the area of a square is very similar.
The area of a rectangle is given by\text{Area} = \text{length} \times \text{width} \text{, or }\\ \\ A = l \times w
The area of a square is given by\text{Area} = \text{side} \times \text{side} \text{, or }\\ \\ A =s \times s=s^2
Examples
Example 1
Find the area of the rectangle shown.
Example 2
Find the area of the square shown.
The area of the rectangle is given by:
The area of the square is given by:
Find dimensions from the area
We have found that the area of a rectangle is given by the product of its length and width. If we already know the area, along with one of the dimensions, we can use this relationship to find the remaining dimension.
Examples
Example 3
Find the width of this rectangle if its area is 66 \text{ m}^2 and its length is 11 m.
We can find a missing dimension of a rectangle by substituting the area and given dimension into the formula for the area of a rectangle: A=l\times w and then solving for the unknown variable.