4.4 G) Similar Shapes with Ratios – Part 1
Sometimes we will be given the ratio of the lengths of the sides of similar shapes. For example, we have two similar shapes (A and B). The ratio of the lengths of the sides of A to the lengths of the sides of B is:
The Areas
From the table at the top of this page, we know that the area scale factor is the length scale factor squared (LSF = n and ASF = n2). Therefore, we work out the ratio of the area of A to the area of B by squaring both of the components in the length ratio. The working is shown below:
The Volumes
I am now going to work out the ratio of the volumes of the two similar shapes. From the table at the top of this page, we know that the volume scale factor is the cube of the length scale factor (LSF = n and VSF = n3). This means that we can work out the ratio for the volume of A to the volume of B by cubing both of the components in the length ratio. The working is shown below:
We are now going to have a look at an example where we are given the area ratio for two similar shapes and we are asked to find the volume ratio for the two similar shapes.
The two shapes, C and D, are mathematically similar. The ratio of the surface area of shape C to the surface area of shape D is:
What is the ratio of the volume of shape C to the volume of shape D?
The question gives us the ratio for the surface areas of the two similar shapes and we want to find out what the ratio is for the volumes of the two similar shapes. We cannot go straight from the ratio of the surface areas to the ratio of the volumes. Instead, we must go from the ratio of the surface areas to the ratio of the lengths. And then, we go from the ratio of the lengths to the ratio of the volumes.
The area scale factor is the length scale factor squared (LSF = n and ASF = n2). This means that we can find the length ratio by square rooting the components in the area ratio. The working is shown below:
We now have the ratio of the lengths of C to the lengths of D.
The volume scale factor is the length scale factor cubed (LSF = n and VSF = n3). This means that we can work out the volume ratio by cubing the components in the length ratio. The working is shown below: