Introduction to Area

If you needed to cover the floor of your house from wall to wall, you would need to share the area of the floor at the carpet store, after which they would tell you how much carpet you would need and what it would cost. If you wanted to varnish the dining room table, you would need to know the area so that you can confirm the cost. These are just a few everyday examples of how areas of closed shapes are important to know and calculate.

The Big Idea: The two dimensions of the area

What is the area? The area of a closed two-dimensional figure (like a circle, triangle, rectangle, square, rhombus or parallelogram) is the amount of space enclosed by the outer boundary of the figure.

This outer boundary is referred to as the perimeter and is a one-dimensional measure of length. But as soon as we try to calculate area, we get into two dimensions of measurement.

There is another measure which is used for closed spaces, and it is called the perimeter, which is a one-dimensional number. It refers to the length of the boundary of the figure. In the example of the varnishing of the table, if you wanted to find out the length of the wooden edge of the table, you would add up all four sides of the rectangular table to get the length of the perimeter.

However, when it comes to area, take a look at the graph illustration below and try to understand why it is that we multiply the sides in order to get the desired result:

The following image is another way of visualising the basic difference between area and perimeter, remember, area represents the space enclosed by a figure, it is two dimensional, while the perimeter is the length of the edge, and is a one-dimensional quantity.

How is it important?

Area of a rectangle

The only difference between a square and a rectangle is that a rectangle has both sets of opposite sides equal, instead of all four sides being equal. So, if we represent the length of a rectangle by the letter ‘$${l}$$’, and the breadth by the letter ‘$${b}$$’, then

\begin{align}{\text{Area of rectangle}} = {\text{length}} \times {\text{breadth}} = l \times b\end{align}

If you have a rectangular wall in your house which needs painting, then you can calculate the area to be painted by seeing what the length and the height of the wall is. For all four walls of a room, you calculate the areas of the two opposite rectangular walls and add them together. If you need the ceiling painted as well, you need to calculate the area of the rectangular ceiling also.

Area of a square

A square is a four-sided figure in which all four sides are equal to each other, and all four angles are right angles. If we represent each side of a square by the letter $${a}$$, then

\begin{align}{\text{Area of square}} = a \times a = {a^2}\end{align}

If we continue the example of the sandwich and consider the original square-shaped bread slices (before you cut them into two triangles), then the area would be calculated as

\begin{align}{\text{Area of each square sandwich}} = 5 \times 5 = {25}\,\rm{cm^2}\end{align}

Area of a triangle

A triangle is a closed space enclosed by three sides. If we keep any side parallel to the bottom of the page, then that side is called the base of the triangle and its length is represented by the letter $${b}$$. The perpendicular line joining the vertex opposite this base to a point in this base is called the height or altitude of the triangle and is usually represented by the letter $${h}$$. The formula for the area of a triangle is then written as:

\begin{align}{\text{Area of triangle}} = \frac{1}{2} \times b \times h\end{align}

Another interesting way to understand the origin of this formula is to picture it as half the area enclosed by a rectangle. To help visualize it, look at the image below:

If you want further clarification on this concept, take a closer look at the next section.

A simple tip

Did you notice that the area of the triangular sandwich was exactly half of the area of the square sandwich? That is because the ‘$${b}$$’ and ‘$${h}$$’ of the triangle were both equal to the ‘$${a}$$’ of the rectangle. Have a look at this image to get a better idea.