8.4 Proportional Relationships Between Similar 2-D and 3-D Figures
Key Takeaways
- If two similar figures have linear scale factor k, their corresponding areas are in ratio k² and their volumes in ratio k³.
- Doubling every linear dimension multiplies area by 4 and volume by 8, which is why large containers hold disproportionately more than small ones.
- Working backward, a known area ratio gives the linear scale factor by taking a square root, and a known volume ratio by taking a cube root.
- Scale drawings and maps apply the linear factor to distances and the squared factor to represented areas.
- Surface area scales as k², the same as any area, so surface-area-to-volume ratio decreases as a figure grows.
8.4 Proportional Relationships Between Similar 2-D and 3-D Figures
Skill 6 of Competency 3 is one idea applied repeatedly, and it is the source of the most persistent misconception in middle-grades geometry: assuming that if lengths double, area doubles too.
The k, k², k³ relationship
If two similar figures have a linear scale factor of k, then:
- corresponding lengths are in ratio k
- corresponding areas (including surface areas) are in ratio k²
- corresponding volumes are in ratio k³
+---------------------------------------------------------------------------+
| Scale factor k | Length x k | Area x k^2 | Volume x k^3 |
+---------------------------------------------------------------------------+
| 2 | 2 | 4 | 8 |
| 3 | 3 | 9 | 27 |
| 1/2 | 1/2 | 1/4 | 1/8 |
| 5 | 5 | 25 | 125 |
| 2/3 | 2/3 | 4/9 | 8/27 |
+---------------------------------------------------------------------------+
The reason is dimensional. Area is a product of two lengths, so scaling each by k scales the product by k². Volume is a product of three lengths, giving k³.
A rectangle 4 cm by 7 cm has area 28 cm². Tripling both dimensions gives 12 cm by 21 cm, area 252 cm². And 252/28 = 9 = 3². ✓
A cube of edge 5 has volume 125. Doubling the edge to 10 gives volume 1,000, and 1,000/125 = 8 = 2³. ✓
Working forward
Two similar triangles have a scale factor of 3 : 5. The smaller has area 45 cm². Find the larger area. Area ratio = (3/5)² = 9/25. So 45/A = 9/25 → 9A = 1,125 → A = 125 cm².
Two similar cylinders have radii 4 and 10. The smaller holds 96 mL. Find the larger capacity. Linear factor k = 10/4 = 2.5, so volume factor = 2.5³ = 15.625. Capacity = 96 × 15.625 = 1,500 mL.
Working backward
Recovering the linear factor from an area or volume ratio requires a root.
Two similar pentagons have areas 48 and 108 cm². Find the ratio of their perimeters. Area ratio = 48/108 = 4/9, so k = √(4/9) = 2/3. The perimeters are in ratio 2 : 3.
Two similar spheres have volumes 54 and 128 cm³. Find the ratio of their radii. Volume ratio = 54/128 = 27/64, so k = ∛(27/64) = 3/4.
Perimeter, circumference, radius, height, and any other length all scale by k — never by k² or k³. Items pair an area ratio with a perimeter question precisely to test whether you take the root.
Scale drawings, maps, and models
A scale of 1 : 200 means every length on the drawing represents 200 of the same units in reality.
A floor plan uses 1 cm : 2.5 m. A room measures 6 cm by 4 cm on the plan. Actual dimensions: 15 m by 10 m, so the actual area is 150 m². The plan area is 24 cm². Note that 150 m² is not 24 × 2.5 = 60; it is 24 × 2.5² = 24 × 6.25 = 150 m². ✓
That check illustrates the trap cleanly: converting a drawn area to a real area uses the squared factor.
A model car is built at 1 : 18 scale. The real car is 4.5 m long and its trunk holds 450 L. Model length = 4.5/18 = 0.25 m. Model trunk capacity = 450/18³ = 450/5,832 ≈ 0.077 L, about 77 mL.
Surface-area-to-volume ratio
Because surface area scales by k² while volume scales by k³, the ratio of surface area to volume scales by 1/k — it decreases as a figure grows.
A cube of edge 2 has surface area 24 and volume 8, a ratio of 3 : 1. A cube of edge 6 has surface area 216 and volume 216, a ratio of 1 : 1.
This explains real phenomena that appear in applied items: small animals lose heat faster relative to their mass, crushed ice melts faster than a single block of the same total volume, and small cells exchange nutrients more efficiently than large ones. Items often frame the mathematics inside a science context, which also serves Competency 5's cross-subject modeling skill.
Similar solids must be genuinely similar
The k³ rule applies only when every dimension is scaled by the same factor. A cylinder whose radius doubles while its height stays fixed is not similar to the original, and its volume multiplies by 4 (from r²), not 8.
Original cylinder r = 3, h = 10: V = 90π. Radius doubled only, r = 6, h = 10: V = 360π, a factor of 4. Both doubled, r = 6, h = 20: V = 720π, a factor of 8. ✓
Distinguishing "all dimensions scaled" from "one dimension scaled" is a genuine reasoning item, and the values 4 and 8 both appear as options.
Two similar triangular banners have a linear scale factor of 2 : 7. If the smaller banner requires 12 square feet of fabric, how much fabric does the larger require?
Two similar cones have volumes of 40 cm³ and 1,080 cm³. What is the ratio of their heights?
A cylindrical tank has its radius doubled while its height stays the same. By what factor does its volume increase?