The sides of three cubes are in the ratio five to six to four. What is the ratio of their lateral surface areas?
The keyword here is “areas.” A quick way to solve this problem would be to recall that an area scale factor is equal to a length scale factor squared. We are told that the side lengths of the cubes are in the ratio five, six, four. This means that all three cubes are in proportion to one another and are similar.
We can therefore find a scale factor that links a cube with side length of one unit to each of the other cubes. We calculate the area of a square by squaring its side length. This means that the ratio of the areas will be five squared to six squared to four squared. These are equal to 25, 36, and 16, respectively. The ratio of the lateral surface areas of three cubes whose sides are in the ratio five, six, four is 25, 36, 16.
A longer way of solving this problem would be to draw the cubes. The first cube has side length five units. This means that the area of one face would be equal to five squared or five multiplied by five. This is equal to 25. As a cube has six faces with equal area, the lateral surface area or total surface area will be equal to six multiplied by 25. This is equal to 150 square units.
We can repeat this process for a cube with side length six units. This would have a lateral surface area of six multiplied by six squared. This is equal to 216 square units.
Finally, a cube with side length four units would have a lateral surface area of six multiplied by four squared. This is equal to 96.
The ratio of the lateral surface areas is therefore 150, 216, 96. All of these values are divisible by six. We might have recognized that as all of the cubes had six faces. Dividing the three values by six gives us 25, 36, and 16. This is the same ratio we found using the first method.