Lesson 8 — Similar Solids
Big idea: When every corresponding dimension of two similar solids is multiplied by r, surface areas are multiplied by r² and volumes by r³.
Words to know
| Word | Meaning |
|---|---|
| Similar solids | Solids in which every pair of distances between corresponding points is proportional. |
| Ratio of corresponding dimensions | The scale factor r: the length of a segment in one solid divided by the corresponding length in the other. |
| Corresponding dimensions | Matching lengths in two similar solids, such as radii, heights, or edge lengths. |
Concepts to know
- If two polyhedra are similar, their corresponding faces are similar, so corresponding dimensions are proportional.
- For two similar rectangular solids, l₁/l₂ = w₁/w₂ = h₁/h₂ = r.
- Theorem 85: If the ratio of a pair of corresponding dimensions of two similar solids is r, then the ratio of their surface areas is r².
- Theorem 86: If the ratio of a pair of corresponding dimensions of two similar solids is r, then the ratio of their volumes is r³.
- These theorems hold for any pair of similar solids, not just rectangular solids or cylinders.
- Area is measured in square units and volume in cubic units, which is why the ratios are squared and cubed.
- Example: a model ship 20 times as long as the real ship has 20² = 400 times the surface area and 20³ = 8,000 times the volume.
Formulas / rules
- Scale factor: r = a₁ ÷ a₂
- Surface area ratio: A₁ ÷ A₂ = r²
- Volume ratio: V₁ ÷ V₂ = r³
- Finding r from volumes: r = ∛(V₁ ÷ V₂)
- Finding r from areas: r = √(A₁ ÷ A₂)
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Compare surface areas | Find r first, then square it. |
| Compare volumes | Find r first, then cube it. |
| Work backward from volumes | Take the cube root of the volume ratio to get r. |
| Work backward from areas | Take the square root of the area ratio to get r. |
| Find a missing dimension | Use the same r for every pair of corresponding lengths. |
| Decide whether solids are similar | Check that all corresponding dimensions have the same ratio. |
Watch out
- Length ratios use r, area ratios use r², and volume ratios use r³.
- Doubling every dimension makes the volume 8 times as large, not 2 times.
- Do not use r² for volume or r³ for area.
- Similar solids must have the same shape; equal volume does not make solids similar.
- Check that you are comparing corresponding parts, not just any two parts.