Mathematics • Year 8 • Unit 3 • Lesson 10
Volume of Prisms — Mixed Challenge
Six mixed volume problems across rectangular, triangular, composite, and trapezoidal prisms. One "find the mistake" and one open-ended container design.
1. Mixed problems — choose the right move
Identify the cross-section, find its area, then multiply by the prism length. Show your working. 3 marks each
1.1 A cube has volume 729 cm³. Find its side length.
1.2 A rectangular box has volume 480 cm³, l = 12 cm, and w = 5 cm. Find h.
1.3 A triangular prism: triangle base 8 cm, triangle height 9 cm, prism length 15 cm. Find V.
1.4 A trapezoidal prism. Cross-section is a trapezium with parallel sides a = 6 cm, b = 10 cm, and height 4 cm. Prism length 12 cm. (Hint: A_trap = ½(a + b) × h.)
1.5 A rectangular prism has dimensions 25 cm × 15 cm × 8 cm. Find its volume in cm³, then convert to litres.
1.6 A swimming pool has a trapezoidal cross-section (shallow end 1 m deep, deep end 2.5 m deep, both ends connected by a sloping floor 10 m long). The pool is 6 m wide. Find the volume of water it holds when full, in m³ and in litres.
2. Find the mistake
A Year 8 student tried to find the volume of a triangular prism with triangle base 6 cm, triangle height 8 cm, and prism length 10 cm. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, and re-do the working. 3 marks
Student's working — find V when b = 6, h_△ = 8, prism length l = 10:
Line 1: V = A_base × h
Line 2: A_triangle = ½ × 6 × 10 = 30 cm²
Line 3: V = 30 × 8 = 240 cm³
Line 4: So the volume is 240 cm³.
(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 4 — "Spot the Trap". The student used the PRISM length (10) as the triangle's height, instead of the actual triangle height (8).3. Open-ended challenge — design a 1 L container
This question has more than one valid answer. 4 marks
3.1 Your task: design THREE different rectangular prism containers, each with a capacity of exactly 1 litre (= 1000 mL = 1000 cm³).
For each container:
(i) State the dimensions l × w × h in cm (use whole numbers).
(ii) Show V = l × w × h = 1000 cm³ exactly.
(iii) Calculate the surface area SA = 2(lw + lh + wh).
Bonus: rank your three containers from smallest SA to largest SA. Which shape uses the LEAST packaging material per litre? Why might a manufacturer prefer that shape?
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Cube V = 729
s³ = 729, so s = ∛729 = 9 cm.
1.2 — Box V = 480, l = 12, w = 5
V = lwh → 480 = 12 × 5 × h → 480 = 60h → h = 8 cm.
1.3 — Triangular prism (8, 9, 15)
A_triangle = ½ × 8 × 9 = 36 cm². V = 36 × 15 = 540 cm³.
1.4 — Trapezoidal prism
A_trap = ½(6 + 10) × 4 = ½ × 16 × 4 = 32 cm². V = 32 × 12 = 384 cm³.
1.5 — Box 25 × 15 × 8
V = 25 × 15 × 8 = 3000 cm³ = 3 L.
1.6 — Swimming pool (trapezoidal cross-section)
A_trap = ½(1 + 2.5) × 10 = ½ × 3.5 × 10 = 17.5 m². V = 17.5 × 6 = 105 m³ = 105 000 L.
2 — Find the mistake
(a) The mistake is on Line 2 (carried into Line 3).
(b) The student used the PRISM length (10 cm) as the triangle's height in the area formula, instead of the triangle's actual height (8 cm). The triangle has base 6 and height 8 — the prism length 10 is separate.
(c) Corrected: A_triangle = ½ × 6 × 8 = 24 cm². V = 24 × 10 = 240 cm³.
(Sanity check: by coincidence the final answer of 240 cm³ is the same here! But the WORKING was wrong: ½ × 6 × 10 × 8 = ½ × 6 × 8 × 10 due to commutativity. The student got the right answer for the wrong reason. The label of which number is the triangle height vs the prism length matters when the numbers differ.)
3 — Design three 1L containers (sample solutions)
Many valid answers. Three examples (all with V = 1000 cm³):
Design A — cube-like: 10 × 10 × 10.
V = 1000 ✓. SA = 2(100 + 100 + 100) = 600 cm².
Design B — short and wide: 20 × 10 × 5.
V = 1000 ✓. SA = 2(200 + 100 + 50) = 2 × 350 = 700 cm².
Design C — long and thin: 25 × 8 × 5.
V = 1000 ✓. SA = 2(200 + 125 + 40) = 2 × 365 = 730 cm².
Ranking (smallest SA → largest): A (600) < B (700) < C (730). The cube-like shape uses LEAST packaging per litre. Manufacturers would prefer the cube because it saves on material costs (less cardboard / plastic / tin per container). However, real packaging is rarely a perfect cube — supermarket shelves are designed for tall, thin boxes (milk cartons) for stacking and display reasons.
Marking: 1 mark per valid design with correct V and SA check (3 marks). 1 mark for ranking and stating cube is most efficient.