Questions: Scale Drawings and Maps

Question 1

A floor plan uses a scale of 1 cm = 4 m. A room measures 3 cm × 5 cm on the plan. What is the room's actual area?

• A. 60 m² — multiply the drawing area (15 cm²) by the scale factor (4)
• B. 240 m² — the actual dimensions are 12 m × 20 m, so area = 240 m²
• C. 15 m² — area in the drawing equals area in reality when scale is applied
• D. 120 m² — multiply the drawing area by the scale factor squared divided by 2

Answer: (B)

The actual dimensions are 3 × 4 = 12 m and 5 × 4 = 20 m, giving an actual area of 12 × 20 = 240 m². The most tempting wrong answer is option A: students often multiply the drawing area (15 cm²) directly by the scale factor (4), getting 60 m². But area scales by the square of the scale factor — k = 4 means area scales by k² = 16. So 15 × 16 = 240 m². Lengths scale linearly; area does not.

Question 2

A scale model of a car is built at 1:20 (every 1 cm on the model = 20 cm on the real car). If it takes 1 can of paint to cover the model, approximately how many cans would be needed to paint the real car's surface?

• A. 20 cans — the real car is 20 times larger, so it needs 20 times as much paint
• B. 40 cans — double the scale factor to account for both dimensions
• C. 400 cans — surface area scales by the square of the scale factor (20² = 400)
• D. 8,000 cans — volume scales by the cube of the scale factor

Answer: (C)

Paint covers surface area, and area scales by k² where k is the linear scale factor. Here k = 20, so area scales by 400. The real car needs 400 times as much paint as the model. Option A (×20) is the classic misconception — applying the linear scale factor to area. The scale factor of 20 applies to each linear dimension; since area = length × width, the scale factor applies twice: 20 × 20 = 400.

Question 3

On a map with scale 1:50,000, a distance of 3 cm represents 1.5 km in reality.

• T. True
• F. False

Answer: True

At 1:50,000, every 1 cm on the map represents 50,000 cm = 500 m = 0.5 km in reality. So 3 cm represents 3 × 0.5 = 1.5 km. This is correct. The proportion is straightforward: 1 cm → 0.5 km, so multiply both sides of the ratio by 3.

Question 4

If a scale drawing is enlarged by a factor of 3 (most lengths become 3 times as long), the total area of the drawing also becomes 3 times as large.

• T. True
• F. False

Answer: False

When lengths scale by a factor of 3, area scales by 3² = 9. A rectangle that is 2 × 4 has area 8; scaled up by 3, it becomes 6 × 12 = area 72 — nine times larger, not three times. Area is length × length, so the scale factor applies twice. Students commonly assume area scales linearly with lengths, but the exponent changes for different dimensions: lengths scale by k, areas by k², volumes by k³.

Question 5

Why doesn't the scale factor for lengths also apply directly to areas? Explain the mathematical reason.

Model answer: Area is calculated by multiplying two lengths together (e.g., length × width). When you scale both lengths by a factor of k, the area becomes (k × length) × (k × width) = k² × (length × width). So the area is multiplied by k², not k. For example, if you double all lengths (k = 2), a 3 m × 4 m room becomes 6 m × 8 m, and the area goes from 12 m² to 48 m² — four times as large, not twice.

This is not a special rule — it follows directly from the definition of area as a product of two lengths. Any time you have a quantity formed by multiplying dimensions together, the scale factor applies once per dimension: lengths scale by k¹, areas (two dimensions) by k², volumes (three dimensions) by k³. Recognizing this pattern prevents the persistent error of applying the linear scale factor to areas.