- 1). Use a calculator to convert both areas to the same units. For example, if you were given two similar boxes that have areas of 2.25 square feet and 144 square inches, you would either convert 2.25 square feet to square inches, or vice versa. Because there are 12 inches in 1 foot, there are 12^2 = 144 square inches in a square foot. Thus, the boxes have areas of 2.25 square feet and 1 square foot, respectively.
- 2). Multiply the numbers obtained in Step 1 by an appropriate integer to form a ratio made of whole numbers. For example, the ratio obtained in the last example was 2.25:1. Multiplying both of those numbers by four, you would get 9:4.
- 3). Take the square root of both parts of the ratio. For example, if the area ratio is 9:4, the square root of that ratio would be 3:2 (the square root of 9 is 3, and the square root of 4 is 2). Because area bears a squared relationship to the side lengths of the solids, taking the square root of their area ratio will yield the ratio of the side lengths of the two objects.
- 4). Cube the ratio obtained in Step 3; that is, raise both numbers to the third power. For example, the ratio obtained in the previous example was 3:2. Cubing both numbers in the ratio would yield 27:8. That is the volume ratio. Because volume bears a cubed relationship to the side lengths of the solids, cubing the ratio of the side lengths will yield the ratio of their volumes.
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