Skip to main content

Dimensional Relationships

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

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"Dimensional Relationships." Mathematics. . 19 Aug. 2018 <>.

"Dimensional Relationships." Mathematics. . (August 19, 2018).

"Dimensional Relationships." Mathematics. . Retrieved August 19, 2018 from

Learn more about citation styles

Citation styles gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, cannot guarantee each citation it generates. Therefore, it’s best to use citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

The Chicago Manual of Style

American Psychological Association

  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.