Mathematics • Year 8 • Unit 3 • Lesson 20
Similarity in the Real World
Use scale factor k where it actually shows up: photo enlargements, model cars, maps, photocopiers and architects' drawings. Then explain why areas scale by k², not k, in your own words.
1. Word problems
Each problem hides a pair of similar figures. Find k first, then apply it (with k for lengths, k² for areas). Show working — a single final answer with no working earns only half marks.
1.1 — Photo enlargement. A photograph is 10 cm wide and 15 cm tall. It is enlarged so the new width is 25 cm.
(a) Find the scale factor k.
(b) Find the new height.
(c) Find the ratio of areas (new : original). 3 marks
1.2 — Model car (1 : 20 scale). A toy model car uses a scale of 1 : 20. The real car is 4.2 m long. The real windscreen has area 0.8 m².
(a) Find the length of the model in cm.
(b) Find the area of the model windscreen in cm².
(c) Explain in one sentence why the area is divided by 400 (not by 20). 3 marks
1.3 — Photocopier shrink-to-fit. A photocopier is set to "75% reduction". A document page has area 600 cm² and perimeter 100 cm.
(a) What is the scale factor k for this reduction?
(b) Find the perimeter of the reduced copy.
(c) Find the area of the reduced copy. 3 marks
1.4 — Map scale. A map of a national park uses a scale of 1 : 50 000. On the map, a hiking trail is 8 cm long, and a campground covers an area of 4 cm².
(a) Find the real length of the trail in km.
(b) Find the real area of the campground in m². (Hint: convert k from cm to m first.)
(c) State whether the linear scale factor and area scale factor are k or k². 3 marks
1.5 — Architect's floor plan. An architect draws a kitchen using a scale of 1 cm = 50 cm (so k = 50, drawing → real). On the drawing the kitchen is 6 cm by 4 cm.
(a) Find the real dimensions of the kitchen, in metres.
(b) Find the real floor area of the kitchen, in m².
(c) Compare: how many times bigger is the real area than the drawing area? Express this as k². 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate says: "If I make a rectangle twice as long and twice as wide, the new area is twice the old area." In your own words, explain (i) why this is wrong, (ii) what the correct multiplier for the area actually is, and (iii) what happens to the perimeter when both sides double. Use a small numerical example to back up your answer (for instance, start with a 3 cm × 4 cm rectangle). Use the phrase "areas scale by k²" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Photo enlargement
(a) k = 25 ÷ 10 = 2.5.
(b) New height = 15 × 2.5 = 37.5 cm.
(c) Area ratio = k² = 2.5² = 6.25, so area ratio = 6.25 : 1 (or 25 : 4).
1.2 — Model car (1 : 20)
(a) Model length = 4.2 m ÷ 20 = 0.21 m = 21 cm.
(b) Area scale factor = (1/20)² = 1/400. Model area = 0.8 m² × 1/400 = 0.002 m² = 0.002 × 10 000 cm² = 20 cm².
(c) Area is "length × length", so when each length is divided by 20, the area is divided by 20 × 20 = 400.
1.3 — 75% reduction
(a) k = 0.75 (a reduction, k < 1).
(b) Reduced perimeter = 100 × 0.75 = 75 cm (perimeter scales by k).
(c) Reduced area = 600 × 0.75² = 600 × 0.5625 = 337.5 cm² (area scales by k²).
1.4 — Map 1 : 50 000
(a) Real length = 8 cm × 50 000 = 400 000 cm = 4000 m = 4 km.
(b) Real area = 4 cm² × 50 000² = 4 × 2 500 000 000 cm² = 1.0 × 10¹⁰ cm² = 1.0 × 10⁶ m² = 1 000 000 m² (i.e. 1 km²).
(c) The linear scale factor is k (used for lengths) and the area scale factor is k² (used for areas).
1.5 — Architect's plan (k = 50)
(a) Real width = 6 × 50 = 300 cm = 3 m. Real length = 4 × 50 = 200 cm = 2 m.
(b) Real floor area = 3 × 2 = 6 m².
(c) Drawing area = 6 × 4 = 24 cm² = 0.0024 m². Ratio = 6 ÷ 0.0024 = 2500. This equals k² = 50² = 2500. ✓
2.1 — Area scaling (sample response)
The classmate is wrong because they have used the linear scale factor k where the area scale factor k² is needed. When both sides of a rectangle double, k = 2 — but areas scale by k², so the new area is 2² = 4 times the old, not 2 times. Numerical check: a 3 cm × 4 cm rectangle has area 12 cm² and perimeter 14 cm. After doubling both sides we get a 6 cm × 8 cm rectangle, with area 48 cm² (which is 4 × 12, matching k² = 4) and perimeter 28 cm (which is 2 × 14, matching k = 2 for perimeter). So perimeter doubles, but area quadruples.
Marking: 1 mark for identifying the linear-vs-area confusion; 1 mark for stating area scales by k² (= 4); 1 mark for the perimeter scaling correctly (× k = 2); 1 mark for a clear numerical example backing it up, with the required phrase used.