Similarity, area and volume scale
Predict how enlarging lengths changes areas and volumes.
Before you begin
- Ratios
- Area and volume formulae
Match corresponding lengths
Similar figures have equal corresponding angles and the same ratio for every pair of corresponding lengths. An enlargement with positive scale factor k multiplies all lengths by k. Match sides using position and angle relationships, not just the order in which numbers appear. A scale factor between zero and one makes a smaller similar figure.
Use the correct power of the scale factor
Area involves two length dimensions, so similar areas scale by k². Volume involves three, so similar volumes scale by k³. This is why doubling a cube's side multiplies its volume by eight. When given an area or volume ratio, work backwards using a square or cube root to find the length ratio.
Worked example
Two similar solids have length ratio small:large = 2:3. The small volume is 80 cm³. Find the large volume.
Show the worked solution
- The small-to-large length multiplier is 3/2.
- Cube it to get a volume multiplier of 27/8.
- Calculate 80 × 27/8 = 270.
270 cm³.
Try it yourself
Similar triangles have length scale factor 4. What is the area factor?
Common mistakes
- Do not multiply a volume by the length scale factor alone.
What can you explain now?
Two similar shapes have areas 25 cm² and 100 cm². Find the length multiplier from small to large.
Compare with the explanation
2.
The area multiplier is 100/25 = 4. Its positive square root gives the length multiplier, √4 = 2.
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Original StudyForA content. AI-assisted checks completed; subject-teacher review is still pending. These lessons are not endorsed by an exam board.
Common Core mathematics standards ↗