Scale Factors for Length, Area, and Volume

14.5: Scale Factors for Length, Area, and Volume. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 5 of 63 parts Practice Flashcards

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Length scaling

Objective: Apply a similarity factor to a perimeter.

Prerequisite skills: similarity and powers. All corresponding lengths scale by k in similar figures, so sums of those lengths, including perimeter and circumference, also scale by k. Ratios of corresponding dimensionless angles stay unchanged. Common error: squaring the scale factor for perimeter. Restrictions and interpretation: Length, area, and volume use scale-factor powers 1, 2, and 3 respectively.

Formulas

$P₂=kP₁$

Worked examples

Try itTwo similar polygons have side-length scale factor 3 from small to large. The smaller perimeter is 33. What is the larger perimeter?Show answer

Each side triples, so their sum triples. Larger perimeter=3(33)=99. Therefore the requested answer is 99.

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Length scaling

Objective: Apply a similarity factor to a perimeter.

Formulas

$P₂=kP₁$

Worked examples

Try itTwo similar polygons have side-length scale factor 3 from small to large. The smaller perimeter is 33. What is the larger perimeter?Show answer

Each side triples, so their sum triples. Larger perimeter=3(33)=99. Therefore the requested answer is 99.

Area scaling

Objective: Square the linear scale factor.

Area contains two length dimensions. Scaling both by k multiplies area by k². Conversely, an area ratio A₂/A₁ gives a positive length factor √(A₂/A₁). Common error: using k instead of k² for areas. Restrictions and interpretation: Length, area, and volume use scale-factor powers 1, 2, and 3 respectively.

Formulas

$A₂=k²A₁$

Worked examples

Try itSimilar triangles have side-length scale factor 3 from first to second. The first has area 27. What is the second area?Show answer

Area multiplier is 3²=9. Thus second area=9(27)=243 square units. Therefore the requested answer is 243.

Volume and reverse scaling

Objective: Take a cube root of a volume ratio to recover a length ratio.

Volume contains three length factors, so similar solids have volume ratio k³. To recover a length from a volume ratio, take the positive cube root before scaling a side. Similarity is essential; equal volume alone does not determine dimensions. Common error: using a volume ratio directly as a length ratio. Restrictions and interpretation: Length, area, and volume use scale-factor powers 1, 2, and 3 respectively. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$V₂/V₁=k³ ⇒ k=∛(V₂/V₁)$

Worked examples

Try itTwo similar solids have volume ratio 27:1, larger to smaller. A smaller corresponding edge is 25. What is the larger edge?Show answer

The length factor is ∛27=3. Multiply the smaller edge by 3: 25×3=75. Therefore the requested answer is 75.

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