Mathematics • Year 8 • Unit 3 • Lesson 6
Parallelograms and Trapezia — Mixed Challenge
Pull everything from Lesson 6 together: forward calculations, reverse rearrangements, composite shapes and slant-vs-height traps. Six mixed problems, one "find the mistake", and one open-ended design challenge.
1. Mixed problems — choose the right move
Each question pulls a different idea from Lesson 6. Decide which formula and which value to use before you write. Show your working. 3 marks each
1.1 Parallelogram: b = 18 cm, h = 11 cm. Find A.
1.2 Trapezium: a = 14 cm, b = 22 cm, h = 9 cm. Find A.
1.3 A parallelogram has base 16 m, slant side 12 m, perpendicular height 9 m. Find A. (Be careful which value you use.)
1.4 A trapezium has area 84 m², parallel sides 6 m and 8 m. Find the perpendicular height h.
1.5 A composite garden bed is made of a rectangle (8 m × 3 m) attached to a trapezium with parallel sides 8 m and 5 m and perpendicular height 4 m. Find the total area.
1.6 A parallelogram has area 144 cm² and perpendicular height 9 cm. Find its base length b.
2. Find the mistake
A Year 8 student has tried to find the area of a trapezium with parallel sides a = 7 cm, b = 13 cm and perpendicular height h = 8 cm. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — find A when a = 7, b = 13, h = 8:
Line 1: A = ½(a + b)h
Line 2: A = ½ × (7 + 13) × 8
Line 3: A = (7 + 13) × 8 = 20 × 8 = 160
Line 4: So the area is 160 cm².
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Revisit lesson § Card 9 — "Forgetting the ½ in the trapezium formula" is one of the most common pitfalls.3. Open-ended challenge — design a 100 m² garden
This question has more than one valid answer. 4 marks
3.1 A landscape architect needs to design a garden bed with total area exactly 100 m². The garden must be made from at least two of the following shapes joined together: rectangle, parallelogram, trapezium.
Your task: Design a garden that meets these rules. You must:
(i) Sketch the garden, labelling every side and perpendicular height (use whole-number metres only).
(ii) Calculate the area of each piece using the correct formula.
(iii) Add the piece areas together and show the total is 100 m².
(iv) State which shapes you used.
Bonus: at least one of your pieces must be a trapezium (not just rectangles or parallelograms).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Parallelogram 18 × 11
A = b × h = 18 × 11 = 198 cm².
1.2 — Trapezium 14, 22, 9
A = ½ × (14 + 22) × 9 = ½ × 36 × 9 = 18 × 9 = 162 cm².
1.3 — Parallelogram with slant trap
Use perpendicular height (9 m), not slant side (12 m). A = b × h = 16 × 9 = 144 m².
1.4 — Find h, trapezium
84 = ½ × (6 + 8) × h = ½ × 14 × h = 7h, so h = 84 ÷ 7 = 12 m. Check: ½ × 14 × 12 = 84 ✓.
1.5 — Composite garden bed
Rectangle: 8 × 3 = 24 m². Trapezium: ½ × (8 + 5) × 4 = ½ × 13 × 4 = 26 m². Total = 24 + 26 = 50 m².
1.6 — Find b, parallelogram
b = A ÷ h = 144 ÷ 9 = 16 cm. Check: 16 × 9 = 144 ✓.
2 — Find the mistake
(a) The mistake is on Line 3 (carried into Line 4).
(b) The student dropped the ½ when moving from Line 2 to Line 3. The trapezium formula is A = ½(a + b)h, so you must halve the product, not just multiply (a + b) by h.
(c) Corrected working:
A = ½(a + b)h = ½ × (7 + 13) × 8 = ½ × 20 × 8 = 10 × 8 = 80 cm². ✓
Sanity check: forgetting the ½ doubles the answer, which is exactly what happened (160 = 2 × 80).
3 — Garden design (sample solution)
There are many valid solutions. One good example:
Design 1: Rectangle 10 m × 6 m + Trapezium with a = 5 m, b = 11 m, h = 5 m.
Rectangle area = 10 × 6 = 60 m². Trapezium area = ½ × (5 + 11) × 5 = ½ × 16 × 5 = 40 m². Total = 60 + 40 = 100 m² ✓.
Design 2: Parallelogram b = 12 m, h = 5 m + Trapezium a = 4 m, b = 6 m, h = 8 m.
Parallelogram = 12 × 5 = 60 m². Trapezium = ½ × (4 + 6) × 8 = ½ × 10 × 8 = 40 m². Total = 100 m² ✓.
Design 3: Rectangle 7 × 8 + Parallelogram b = 11, h = 4.
56 + 44 = 100 m² ✓ (but no trapezium — bonus not earned).
Marking: 1 mark for sketch with labelled measurements; 1 mark for at least two shapes used; 1 mark for correct individual area calculations; 1 mark for total = 100 m² verified. Bonus mark if a trapezium is included.