Similar Shapes: Area and Volume Scale Factors
Use the fact that areas scale by the square of the length scale factor and volumes by its cube, and work the relationship in both directions.
What a learner can do afterwards
- Find the area of an enlarged shape from the length scale factor
- Work back to the length scale factor from a ratio of volumes
- Explain why doubling every length of a solid multiplies its volume by eight
1 · Read
Start with the length scale factor k, new length divided by old length. Areas scale by k squared, since area multiplies two lengths. A photo enlarged with k equal to 3 grows 9 times in area. A 4 cm by 5 cm photo has area 20, and its enlargement has 20 times 9, which is 180.
Volumes scale by k cubed, since volume multiplies three lengths. Double every edge and the volume grows 8 times, not twice. Two jugs 10 cm and 15 cm tall give k equal to 1.5, and 1.5 cubed is 3.375, so the taller holds over three times as much.
Ratios also run backwards. From an area ratio take the square root, and from a volume ratio take the cube root. Volumes in ratio 27 to 1 mean lengths in ratio 3 to 1. Areas of 20 and 180 mean a length factor of 3.
The power matches the dimension: 1 for length, 2 for area, 3 for volume. Learn k, k squared and k cubed once, then reuse them for every shape. Scaling treats pyramids and boxes alike, so one rule covers them all.
Square k for areas, cube it for volumes, and take roots on the way back.
More to read
How to Find the Scale Factor (With Example Problems)
Start with the length scale factor k: new length divided by old length. Areas then scale by k squared and volumes by k cubed, because area multiplies two lengths and volume multiplies three. A photo enlarged with k = 3 grows 9 times in area, and a solid doubled in every length holds 8 times the volume.
The reverse trip uses roots: from an area ratio take the square root, from a volume ratio the cube root. Volumes in ratio 8:1 mean lengths in ratio 2:1. Learn both directions and you can answer any enlargement question from a single ratio.
Volume formulas review
Volume measures cubes that fill a solid, so scaling lengths hits volume three times over. Doubling every edge of a box multiplies its volume by 2 cubed, which is 8. Tripling multiplies by 27. The exponent matches the dimension: 1 for length, 2 for area, 3 for volume.
For the two jugs 10 cm and 15 cm tall, k is 1.5 and the volume ratio is 1.5 cubed, which is 3.375. The taller jug holds over three times as much, a result that surprises anyone who guesses 1.5.
Volume of a pyramid or cone
Pyramids and cones take one third of their prism or cylinder partners, but scaling treats them identically. Enlarge any solid with length factor k and the volume still multiplies by k cubed, whatever the shape formula. Shape changes the base volume, scale changes it by the cube.
This uniformity is what makes scale reasoning powerful: learn k, k squared, k cubed once and reuse them everywhere. A cone doubled in height and radius alike still grows exactly 8 times in volume, just like a box would.
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2 · Watch
به انگلیسیTake it off screen
Where it sits
Learn first
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
13 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.