4.4 E) Scale Factors: Length, Area & Volume – Part 1
We have two cubes; a smaller cube and a larger cube. The smaller cube has sides of 2 cm and the larger cube has sides of 8 cm.
Let’s first work out the length scale factor. We are going to do this by dividing a side in the larger shape by the corresponding side in the smaller shape (as this is a cube, all of the sides are going to be the same, so any side will be 2 cm in the smaller shape and 8 cm in the larger shape).
According to the rule, the area scale factor will be the length scale factor squared. This means that the area scale factor should be 16 (42 = 4 x 4 = 16). Let’s just check that this is true by working out the surface area for the smaller and larger cube. A cube has 6 faces and the area of each of these faces are the same.
The smaller cube has sides of length 2 cm, which means that the area of one of the faces is 4 cm2 (2 x 2). As there are 6 faces with the same area, we must multiply this value by 6, which tells us that the surface area of the smaller cube is 24 cm2.
The larger cube has sides of length of 8 cm. This means that the area of one of the faces is 64 cm2 (8 x 8). To obtain the overall surface area, we multiply the area of one face by 6 and this gives us a total surface area of 384 cm2 (6 x 64).
We now have both surface areas which means that we can work out the area scale factor. We do this by dividing the surface area of the larger cube by the surface area of the smaller cube (the larger cube has a surface are of 384 cm2 and the smaller cube has a surface area of 24 cm2).
This is exactly the same value as the rule stated that we should obtain.
According to the rule, the volume scale factor should be the length scale factor cubed. Earlier we found that the length scale factor was 4, which means that the volume scale factor should be 64 (43 = 4 x 4 x 4). Let’s check that it is 64 by working out the volumes of the cubes that we were given.
We work out the volume of a cube by cubing one of the sides. The sides of the smaller cube are 2 cm, which means that the smaller shape has a volume of 8 cm3 (23 = 2 x 2 x 2). The larger cube has sides of 8 cm, which means that the volume of the larger cube is 512 cm3 (83 = 8 x 8 x 8).
We can find the volume scale factor by dividing the volume of the larger cube by the volume of the smaller cube. The volume of the larger cube is 512 cm3 and the volume of the smaller cube is 8 cm3.
This gives us exactly the same value as the rule stated.
Therefore, we can see that all of the rules that were given in the table for the three different scale factors work. It is probably worth writing these rules on a revision card.
All Together
The LSF, ASF and VSF for this example are shown in the table below.
Therefore, we can see that all of the rules that were given in the table for the three different scale factors work. It is probably worth writing these rules on a revision card.