4.4 H) Similar Shapes with Ratios – Part 2
We will look at two examples in this section. The first example will solely involve ratios, and the second example will involve ratios and values for the length, area or volume.
The diagram below is made up of 3 mathematically similar shapes.
The ratios of the heights of the shapes are 1 : 3 : 6.
The question gives us the ratio of the heights of the three similar shapes. Height is a length, which means that the question is giving us the length ratio. The length ratio is shown below:
The question wants us to find a ratio to do with areas and this means that we will need to find the area ratio for the three different shapes. We are able to find the area ratio from the length ratio by squaring all of the components in the length ratio. We square the components of the length ratio because the area scale factor is the square of the length scale factor (LSF = n and ASF = n2). The working for finding the area ratio is shown below:
I am going to use the area ratio to work out the shaded part of the shape first. The shaded part of the shape is the area of the middle shape minus the area of the smallest shape. According to the area ratio, the area of the middle shape is 9 and the area of the smallest shape is 1. This means that the area that is shaded is 8 (9 – 1).
This example is a little bit tricky, so you may want to go through this example a couple of times.
The two prisms below are mathematically similar. The volume of prism A is 486 cm3 and the length of prism B is 24 cm.
Find the area of the cross section for B.
This question is a prism question. Therefore, it may be a good idea to write down the formula for working out the volume of a prism. The formula is shown below:
Also, this is a similar shapes question and we are given/ asked to find information about length, area and volume. Therefore, I think that it is a good idea to find the length, area and volume ratios for the similar shapes. The question has already given us the area ratio because we were told that the ratio of the area of the cross section for A to the area of the cross section for B is 9 : 16. We can use this area ratio to work out the length ratio and volume ratio.
We work out the length ratio from the area ratio by square rooting the components in the area ratio. We square root because the area scale factor is n2 and the length scale factor is n. The working for finding the length ratio from the area ratio is shown below.
The length scale factor is n and the volume scale factor is n3. Therefore, we find the ratio of the volumes by cubing the components in the length ratio.
We now have the length ratio (3 : 4), area ratio (9 : 16) and volume ratio (27 : 64), which is handy just in case we need them.
We were told in the question that the volume of A is 486 cm3. We can use this information along with the volume ratio to work out the volume of B (the ratio for the volume of A to the volume of B is 27 : 64). In the volume ratio, the volume of A is represented by 27 parts, and these 27 parts are equal to the volume of A, which is 486 cm3. We can create the following equation from this information:
We can find what 1 part in the ratio is by dividing both sides of the equation by 27.
This tells us that 1 part in the ratio is 18 cm3.
We can now find the volume of B by multiplying the number of parts that represents the volume of B in the volume ratio (64) by what 1 part in the ratio represents (18 cm3). The calculation for the volume of B is shown below:
This tells us that the volume of B is 1,152 cm3.
We now have the volume of B (1,152 cm3) and the length of B (24 cm; this was given to us in the question). We can sub these values into the volume of a prism formula to work out the area of the cross section of B. The formula for the volume of a prism is shown below.
We now sub the volume of the prism in as 1,152 cm3 and the length as 24 cm.
We want to find the area of the cross section, and we are able to do this by dividing both sides of the equation by 24.
This tells us that the area of the cross section of B is 48 cm2.
Another way that we could have answered this question is by using the length ratio to work out the length of A from the length of B. We then could have found the area of the cross section of A by dividing the volume of A by the length of A that we have just found. We would then use the area of the cross section of A and the ratio of areas to work out the area of the cross section of B.