10.6.2 Formulas for Areas

If you know the formula for the area of a rectangle, you can work out formulas for the area of a parallelogram, shown below, and a triangle.

Take a parallelogram, cut off the left edge, and place it next to the right edge to make a rectangle. This has the same area as the parallelogram.

The area of the rectangle can be found by multiplying its base by its height.

area of a parallelogram equals base multiplication height

Now if you cut this parallelogram in half along a diagonal, there are two possibilities:

In each case, the parallelogram has been split into two triangles that have the same area. So the area of each triangle is half the area of the parallelogram.

This gives a general formula for the area of a triangle:

area of a triangle equation left hand side equals right hand side one divided by two postfix multiplication base multiplication height

The height goes through the vertex that is opposite the base and is always perpendicular to the base, as shown in the diagram above.

To find the area of a triangle quickly, you can work out half the base and multiply it by the height (or vice versa). Or you can multiply the base by the height and then divide by 2.

The diagram below is a rough sketch of the gable end of a house that needs weatherproofing.

To work out the quantity of materials required, the area of the wall is needed. We can break this problem down by splitting the area into a rectangle and a triangle, then working out these areas and finally adding the two areas together to get the total. Assume the measurements have been made to the nearest 10 cm.

area of rectangular part equation left hand side equals right hand side length multiplication width equation left hand side equals right hand side prefix of eight m prefix multiplication of six m equation left hand side equals right hand side 48 times m super two

The triangle has a base of length 8 m and a perpendicular height of 2 m. So, the area can be calculated as follows:

area of triangular part equation left hand side equals right hand side one divided by two postfix multiplication base multiplication height equation left hand side equals right hand side one divided by two times eight m prefix multiplication of two m equation left hand side equals right hand side eight times m super two

That means the total area of the gable end = equation left hand side 48 times m super two plus eight times m super two equals right hand side 56 times m super two.

Go to the next page to have a go at some examples for yourself.

10.6.1 Areas of Rectangles

10.6.3 Tangram Areas Again