Dimensional Relationships

Usually, when mathematicians compare the size of two-dimensional objects, they compare their areas. For example, how many times larger is a larger square than a smaller one? One way to answer this question is to determine the lengths of the sides of the squares, and use this information to find the respective areas.

Use the formula for the area of a square, A = S ², where A represents area and S represents the side length of the square. Suppose two squares have side lengths of 2 and 6, respectively. Hence, the respective areas are 4 and 36. Thus the area of the larger square is nine times that of the smaller square. Therefore, a square whose side length is three times that of a second square will have an area nine times as great.

Use the notation S ₁ to denote the side of the smaller square and S ₂ to denote the side of the larger square. With this notation, S ₂ = 3S ₁. The area of the larger square then becomes (3S ₁)² = 3S ₁ × 3S ₁ = 9S ₁². This can be generalized further by letting one side of the square be k times the side of another, also known as the ratio of similitude (k ) between the figures. Then (kS ₁)² = kS ₁ × kS ₁ = k ²S ₁². From this, it is evident that if the side lengths of one square are k times the side lengths of another, the area of the first is k ² that of the other.

This principle is true for any two-dimensional object. Suppose two circles have radii that are in the ratio of 2:1. Letting R ₂ = 2R ₁, the area of the larger circle can be represented by A = π (2R ₁)² = 4πR ₁².

As another example, suppose the sides and altitude of the larger triangle are twice those of a smaller triangle. Thus the area of the larger triangle can be written as .

For three-dimensional objects, volumes of similar figures relate to each other in a manner akin to areas of two-dimensional figures. A cube, for example, with a side length twice that of another cube, will have a volume 2³ = 8 times as great. A sphere with a radius five times that of a smaller sphere will have a volume 5³ = 125 times as great.

If k represents the ratio of similitude of two similar objects, then the areas of the two objects will be in the ratio of k ², and the volumes of the two objects will be in the ratio of k ³.

see also Dimensions.

Albert Goetz