Area and Perimeter Under Scaling
A class designs rectangular garden plots on grid paper. Plot A is 4 cm by 9 cm; Plot B is 8 cm by 18 cm (each side doubled). A student, Tariq, says "B is twice as big as A because every side doubled, so it holds twice as much soil." Another student, Grace, says "The perimeter doubled but the area quadrupled, so B needs four times the fencing and four times the soil." The teacher wants students to separate linear, area, and (for a 3-D extension) volume scaling, and to catch each error precisely.
1. Compute perimeter and area for both plots and evaluate Tariq's and Grace's claims.
Read the worked answer
Plot A (4×9): perimeter = 2(4+9) = 26 cm; area = 4×9 = 36 cm². Plot B (8×18): perimeter = 2(8+18) = 52 cm; area = 8×18 = 144 cm².
- Perimeter: 52/26 = 2 → doubled. (Linear measure scales by the length factor, k = 2.)
- Area: 144/36 = 4 → quadrupled. (Area scales by k² = 2² = 4.)
Tariq is wrong: doubling each side multiplies area by 4, not 2, so B holds four times the soil, not twice. Grace is partly right and partly wrong: she is correct that area quadrupled (so four times the soil). She is incorrect that fencing quadruples-fencing is a perimeter (linear) measure, and perimeter only doubled. She conflated the area factor with the perimeter factor.
2. Generalize the scaling rule and apply it to a 3-D planter that doubles all three dimensions.
Read the worked answer
Scaling rule: if every linear dimension is multiplied by k, then
- lengths/perimeters scale by k,
- areas/surface areas scale by k²,
- volumes scale by k³.
For the 3-D planter with k = 2: volume scales by 2³ = 8, so it holds 8 times as much soil; surface area scales by 2² = 4; edge lengths/perimeter scale by 2.
A way to see it: volume = L×W×H, and each factor doubles, so the product picks up 2×2×2 = 8. The same reasoning gives area = L×W picking up 2×2 = 4. This is why a doubled model "looks" modestly bigger but is dramatically heavier/fuller-volume grows fastest.