Mathematics • Year 8 • Unit 3 • Lesson 5
Area in the Real World
Use A = lw and A = ½bh where area actually shows up: a painted wall, a sail, a backyard lawn, a triangular flag and a composite L-shaped floor. Then explain your thinking in your own words.
1. Word problems
Each problem uses area for something practical — painting, sailing, mowing, decorating. Sketch where useful and show full working. Final answer alone earns only half marks.
1.1 — Painting a wall. A painter charges $18 per m² to paint a wall. The wall is 4.5 m wide and 3 m tall.
(a) Calculate the area of the wall.
(b) Calculate the cost of painting.
(c) Now suppose a 1 m × 2 m window in the wall is NOT painted. Recalculate the cost. 3 marks
1.2 — Triangular sail. A small sailboat has a triangular sail with base 2.4 m along the boom and perpendicular height 3.5 m up the mast.
(a) Sketch the sail with base and height labelled.
(b) Calculate the area of the sail. 3 marks
1.3 — Backyard lawn. Maria's rectangular lawn is 14 m long and 9 m wide. Lawn fertiliser is sold in bags that cover 50 m² each. How many bags should she buy?
(a) Calculate the area of the lawn.
(b) Calculate how many bags are needed (round UP to a whole number — you can't buy half a bag). 3 marks
1.4 — Triangular flag. A right-angled triangular flag has legs of 60 cm and 80 cm. Find its area in cm² and convert to m². (Hint: 10 000 cm² = 1 m².)
3 marks
1.5 — Tiling an L-shaped floor. A kitchen floor is L-shaped: it can be split into a 6 m × 4 m rectangle and a 3 m × 2 m rectangle joined at one edge. Tiles cost $35 per m².
(a) Sketch the floor and find the area of each rectangle.
(b) Find the total floor area.
(c) Calculate the tiling cost. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate is asked to find the area of a triangle with base 8 cm and slant side 10 cm (no perpendicular height is given). They write "A = ½ × 8 × 10 = 40 cm". In your own words, explain (i) the TWO mistakes they have made (one about which side is "h", one about units), (ii) what extra information they would need to actually calculate the area, and (iii) what they should write if the perpendicular height was 6 cm instead. Use the phrase "h must be perpendicular to b" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Painting a wall
(a) A = 4.5 × 3 = 13.5 m².
(b) Cost = 13.5 × $18 = $243.
(c) Window = 1 × 2 = 2 m². Painted area = 13.5 − 2 = 11.5 m². Cost = 11.5 × $18 = $207.
1.2 — Triangular sail
(a) Right triangle: base 2.4 m along boom (horizontal), height 3.5 m up mast (perpendicular).
(b) A = ½ × 2.4 × 3.5 = ½ × 8.4 = 4.2 m².
1.3 — Backyard lawn
(a) A = 14 × 9 = 126 m².
(b) Bags = 126 ÷ 50 = 2.52, round up to 3 bags (you can't buy 2.52 bags).
1.4 — Triangular flag
A = ½ × 60 × 80 = ½ × 4800 = 2400 cm². Convert: 2400 ÷ 10 000 = 0.24 m².
1.5 — Tiling the L-shaped kitchen
(a) Rect 1: 6 × 4 = 24 m². Rect 2: 3 × 2 = 6 m².
(b) Total = 24 + 6 = 30 m².
(c) Cost = 30 × $35 = $1050.
2.1 — Explain your thinking (sample response)
The classmate has made two mistakes. First, they used the slant side (10 cm) as the height, but in the formula A = ½bh, h must be perpendicular to b — that is, at exactly 90° to the base, NOT along the slanted edge of the triangle. Second, they wrote the answer as "cm" instead of "cm²"; area is a 2D measurement, so the units MUST be squared. To actually calculate the area, they need to be given the perpendicular height — the vertical distance from the base to the opposite vertex. If h = 6 cm, then A = ½ × 8 × 6 = 24 cm² (not 40 cm or 40 cm²).
Marking: 1 mark for spotting the "used slant instead of perpendicular height" mistake; 1 mark for spotting the units mistake (cm vs cm²); 1 mark for the corrected calculation with h = 6 → 24 cm²; 1 mark for clear sentences using "h must be perpendicular to b".