4.4 F) Scale Factors: Length, Area & Volume – Part 2
Before we start going through the examples in this section, it is a good idea to remind ourselves of the length, area and volume scale factor rules. These are given in the table below.
We have two triangles that are mathematically similar. The smaller triangle is called A and the larger triangle is called B. Triangle A has an area of 6 cm2 and triangle B has an area of 24 cm2. The base of triangle A is 3. What is the base of shape B (the side that we are looking for is labelled as x on the diagram below).
We are given the areas of the two triangles and from this information, we are able to work out the area scale factor. We can do this by dividing the area of B (the larger triangle) by the area of A (the smaller triangle). The larger triangle has an area of 24 cm2 and the smaller triangle has an area of 6 cm2.
The question asks us to find the length of the base of triangle B. We are going to be using the length of the base in triangle A and the length scale factor to find out what the length of the base in triangle B is. However, at the moment we do not know what the length scale factor is, but, we are able to use the area scale factor that we have found to work out what the length scale factor is. We learnt from the previous section, that the area scale factor is the square of the length scale factor (LSF = n, ASF = n2). Therefore, we can obtain the length scale factor from the area scale factor by square rooting the area scale factor. The area scale factor is 4.
Therefore, the length scale factor is 2.
We are now able to find the base of shape B (the value of x). We are finding the length of a side on the larger shape and this means that we multiply the corresponding side on the smaller shape by the length scale factor. The corresponding side on the smaller shape is 3 cm and the length scale factor is 2. The calculation to find the value of x is shown below.
The length of the base in B is 6 cm (x = 6).
There are two cuboids that are mathematically similar. The smaller cuboid has a volume of 16 cm3 and the larger cuboid has a volume of 432 cm3. The larger cuboid has a surface area of 360 cm2. What is the surface area of the smaller cuboid?
The question is asking us to find the surface area of the smaller cuboid. In order to find the surface area of the smaller shape, we need to find out what the area scale factor is. We are able to find out what the area scale factor is by finding the volume scale factors and using the rules that relate the length scale factor, area scale factor and volume scale factor.
We find the volume scale factor by dividing the volume of the larger cuboid by the volume of the smaller cuboid. The larger cuboid has a volume of 432 cm3 and the smaller cuboid has a volume of 16 cm3. Therefore, the volume scale factor is:
We now have the volume scale factor, which we can use to find the area scale factor. The rules are given in the table below.
From the table, we can see that the area scale factor is equal to the length scale factor squared and the volume scale factor is equal to the length scale factor cubed. Therefore, we are able to find the area scale factor, by using the volume scale factor to find the length scale factor. We then use the length scale factor to find the area scale factor.
We have found that the volume scale factor is 27 and we know that this is equal to the cube of the length scale factor. Therefore, we can find the length scale factor by cube rooting the volume scale factor.
The length scale factor is 3.
The area scale factor (which is what we are looking for) is the length scale factor squared. This means that we find the area scale factor by squaring 3.
The area scale factor is 9. We want to find the area of the smaller shape. This means that we divide the area of the larger shape by the area scale factor. The area of the larger shape is 360 cm2 and the area scale factor is 9. Therefore, we undertake the following calculations:
The surface area of the smaller cuboid is 40 cm2.
Multiplying or Dividing?
If we are finding the length, area or volume of a larger shape from a smaller shape, we will always be multiplying by the appropriate scale factor. If we are finding the length, area or volume of a smaller shape from larger shape, we will be dividing by the appropriate scale factor.
Also, here is a reminder of the three different scale factors.