SAT® · Math: Geometry and Trigonometry
Scale Factor: Lengths, Areas, and Volumes
When a shape is scaled by a factor k, every length multiplies by k, every area multiplies by k2, and every volume multiplies by k3. The extra powers come from area and volume each being built out of two or three lengths multiplied together.
Scale factor describes how a figure grows or shrinks when every length is multiplied by the same number k. If k is the ratio of corresponding sides, then area ratio is k2 and volume ratio is k3. The reason is simple multiplication: area comes from multiplying two lengths together, so each one contributes a factor of k, giving k times k = k2. Volume comes from three lengths multiplied together, so it picks up k times k times k = k3. A box with sides doubled has every face area four times as big, and the whole box holds eight times as much, because 23 = 8.
This shows up constantly in similar figures and solids. If two triangles are similar with side ratio k, their area ratio is k2 - so tripling every side of a triangle multiplies area by 32 = 9. Given an original area of 7, the new area is 7 times 9 = 63. The same logic runs backward with volume: two similar cylinders with volumes 27 and 125 have a volume ratio of 27/125, and taking the cube root gives the length scale factor, 3/5, since (3/5)3 = 27/125.
This idea sits next to area and volume formulas and to right-triangle similarity, since it is really a shortcut for comparing similar shapes without recomputing every formula from scratch. It also connects to unit conversion, where converting cubic units uses the same cube-of-the-ratio reasoning.
Key terms
- Scale factor (k)
- The number every length is multiplied by when a figure is enlarged or shrunk to a similar one.
- Similar figures
- Shapes with the same angles and proportional corresponding sides.
- Area scaling
- Area multiplies by k2 because it is built from two lengths each carrying a factor of k.
- Volume scaling
- Volume multiplies by k3 because it is built from three lengths each carrying a factor of k.
Watch it explained
- How it worksHow does one factor of k per dimension become k squared for area and k cubed for volume?
- Worked exampleFind the area of a triangle after every side is multiplied by 3, given the original area is 7.
- Worked exampleFind the scale factor between two similar cylinders whose volumes are 27 and 125 cubic centimeters.
- Common mistakeWhy doesn't tripling a photo's width and height triple the ink it takes to print?
- Going furtherWhat happens to skin and capacity together when a cube is enlarged by a factor of 10?
Check yourself
End of topic.