Mathematics • Year 8 • Unit 3 • Lesson 6
Parallelograms and Trapezia in the Real World
Use A = bh and A = ½(a + b)h where they actually show up: solar farms, garden beds, kite sails, lawn covers and patios. Then explain your thinking in your own words.
1. Word problems
Each problem hides a parallelogram or a trapezium. Sketch the shape, label sides clearly, then apply the right formula. A single final answer with no working earns only half marks.
1.1 — Solar farm. A solar farm sits on a trapezoidal field with parallel sides 80 m and 120 m, and perpendicular height 60 m between them.
(a) Find the area of the field.
(b) Each solar panel covers 2 m². How many panels can fit on the field?
(c) If each panel produces 0.4 kW, what is the total power produced (kW)? 3 marks
1.2 — Parallelogram garden bed. A garden bed in the shape of a parallelogram has base 4.5 m, slant side 3.2 m, and perpendicular height 2.8 m. A bag of mulch covers 3 m².
(a) Find the area of the bed.
(b) How many bags of mulch are needed (round up)?
(c) Why is the slant side (3.2 m) irrelevant for the area? 3 marks
1.3 — Kite sail. A trapezoidal kite sail has its two parallel edges measuring 1.2 m and 0.6 m, with a perpendicular height of 1.5 m between them.
(a) Find the area of fabric needed for the sail.
(b) Fabric costs $18/m². Find the total cost. 2 marks
1.4 — Lawn cover. A parallelogram-shaped lawn has perpendicular height 7 m. Its area is 91 m².
(a) Find the length of its base (use h = A ÷ b rearranged).
(b) A roll of turf is 1.5 m wide and 10 m long. How many full rolls are needed (round up)? 3 marks
1.5 — Patio paving. A patio is in the shape of a trapezium with parallel sides 6 m and 4 m and perpendicular height 3 m. Pavers come in squares of side 0.5 m (area 0.25 m² each).
(a) Find the patio area.
(b) How many pavers are needed?
(c) Pavers cost $4.20 each. What is the total paving 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 parallelogram with base 9 cm, slant side 7 cm and perpendicular height 5 cm. They write "A = 9 × 7 = 63 cm²". In your own words, explain (i) which value they should have used and which they used by mistake, (ii) the correct answer with working, and (iii) what visual feature on a diagram would have warned them. Use the phrase "perpendicular height, not slant side" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Solar farm
(a) A = ½ × (80 + 120) × 60 = ½ × 200 × 60 = 100 × 60 = 6 000 m².
(b) Panels = 6 000 ÷ 2 = 3 000 panels.
(c) Power = 3 000 × 0.4 = 1 200 kW (1.2 MW).
1.2 — Parallelogram garden bed
(a) A = b × h = 4.5 × 2.8 = 12.6 m².
(b) Bags = 12.6 ÷ 3 = 4.2 → round up to 5 bags.
(c) The slant side (3.2 m) is the length of the leaning edge, not the perpendicular distance between parallel sides. Only the perpendicular height (2.8 m) appears in A = bh.
1.3 — Kite sail
(a) A = ½ × (1.2 + 0.6) × 1.5 = ½ × 1.8 × 1.5 = 0.9 × 1.5 = 1.35 m².
(b) Cost = 1.35 × $18 = $24.30.
1.4 — Lawn cover
(a) b = A ÷ h = 91 ÷ 7 = 13 m.
(b) One roll covers 1.5 × 10 = 15 m². Rolls needed = 91 ÷ 15 = 6.07 → round up to 7 rolls.
1.5 — Patio paving
(a) A = ½ × (6 + 4) × 3 = ½ × 10 × 3 = 15 m².
(b) Pavers = 15 ÷ 0.25 = 60 pavers.
(c) Cost = 60 × $4.20 = $252.
2.1 — Explain your thinking (sample response)
The classmate has used the slant side (7 cm) by mistake, when they should have used the perpendicular height, not slant side. The correct working is A = b × h = 9 × 5 = 45 cm². They could have spotted the mistake by looking for the small right-angle square at the foot of the height line — that square always marks the perpendicular distance between the parallel sides, and that is the value that goes into the formula. The slant side (7 cm) is the length of the leaning edge; it is always longer than the height and never appears in A = bh.
Marking: 1 mark for spotting the "used slant instead of height" mistake; 1 mark for the correct answer 45 cm² with working; 1 mark for the diagram tip (right-angle marker / dashed line / "h" label); 1 mark for clear full-sentence explanation using "perpendicular height, not slant side".