Mathematics • Year 10 • Unit 1 • Lesson 19
Composite Volumes — Mixed Challenge
Pull together every idea from Lesson 19: the addition strategy, the subtraction strategy, Pythagoras to find a missing length, mixed units (cm and m in one diagram), L-shapes and "hole through a block" problems. Choose the right tool for each problem, spot another student's overlap blunder, then invent a composite that hides a Pythagoras step.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 19. Decide whether to use the addition strategy or the subtraction strategy before you start writing. Give exact answers in π unless told otherwise. 3 marks each
1.1 A rectangular block 12 cm × 8 cm × 5 cm has a cylinder of radius 1.5 cm and height 5 cm placed on top. Find the total volume in exact form.
1.2 A cube of side 12 cm has a square hole of side 4 cm cut straight through one face to the opposite face (a 4 cm square × 12 cm long tunnel). Find the remaining volume.
1.3 A right-angled triangular prism has a base triangle with legs 6 cm and 8 cm (hypotenuse 10 cm). The prism length is 12 cm. Find its volume.
1.4 An L-shaped prism has a cross-section made of two rectangles: 8 cm × 5 cm and 5 cm × 3 cm joined along the 5 cm side. The prism is 10 cm long. Find its volume.
1.5 A storage tank is a closed cylinder of radius 1 m and height 3 m, with a half-cylinder "dome" of radius 1 m and length 3 m placed lying on top (so the dome's curved side faces up). Find the total volume in exact form and to 2 d.p.
1.6 A concrete pipe is a cylinder of outer diameter 60 cm, inner diameter 50 cm, and length 2 m. Find the volume of concrete in m³ to 4 d.p.
2. Find the mistake
Another Year 10 student has tried to find the volume of an L-shaped prism using the addition strategy. The L-shape cross-section is made of a 6 cm × 4 cm rectangle and a 4 cm × 3 cm rectangle joined along the common 4 cm side. The prism is 10 cm long. 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 the volume of the L-shaped prism:
Line 1: Rectangle 1: 6 × 4 = 24 cm²
Line 2: Rectangle 2: 4 × 3 = 12 cm²
Line 3: Cross-section area = 24 + 12 = 36 cm²; the overlap is the 4 × 4 = 16 cm² square so I subtract it: 36 − 16 = 20 cm²
Line 4: V = 20 × 10 = 200 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? The two rectangles are joined along the 4 cm edge, they do not overlap. There is no shared area to subtract.3. Open-ended challenge — invent a Pythagoras problem
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Lesson 19 says: "Use Pythagoras to find missing dimensions for volume calculations." Design a composite-solid volume problem where a student must use Pythagoras at least once before they can find the volume.
Your problem must include:
(a) A clear word-problem description (3-4 sentences) of a real-world object made from two or more simple solids.
(b) A 2D sketch (rough is fine) with all the dimensions you give and one missing dimension labelled with a "?".
(c) A solution that uses Pythagoras to find the "?", then computes the total volume in exact form (π OK) and as a decimal to 1 d.p.
Constraint: the Pythagoras step must produce an integer side length (e.g. a 3-4-5, 5-12-13, 6-8-10, 8-15-17 right triangle) so the rest of the working stays clean.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Block 12 × 8 × 5 + cylinder r=1.5, h=5
Vblock = 480 cm³.
Vcyl = π(1.5)²(5) = 11.25π cm³.
Total = 480 + 11.25π cm³ (≈ 515.3 cm³).
1.2 — Cube 12 with 4 × 4 × 12 tunnel
Vcube = 12³ = 1728 cm³.
Vhole = 4 × 4 × 12 = 192 cm³.
Remaining = 1728 − 192 = 1536 cm³.
1.3 — Right-angled triangular prism (legs 6, 8; length 12)
Abase = ½ × 6 × 8 = 24 cm². V = 24 × 12 = 288 cm³.
1.4 — L-shaped prism (8×5 + 5×3; length 10)
Across-section = 8×5 + 5×3 = 40 + 15 = 55 cm². V = 55 × 10 = 550 cm³.
(The two rectangles share an edge, not an area — no overlap to subtract.)
1.5 — Closed cylinder r=1, h=3 + half-cylinder dome r=1, L=3 on top
Vcyl = π(1)²(3) = 3π m³.
Vdome = ½ × π(1)²(3) = 1.5π m³.
Total = 4.5π ≈ 14.14 m³ (2 d.p.).
1.6 — Concrete pipe (outer d = 60 cm, inner d = 50 cm, L = 2 m)
Outer r = 0.30 m, inner r = 0.25 m, L = 2 m.
Vouter = π(0.30)²(2) = 0.18π m³.
Vinner = π(0.25)²(2) = 0.125π m³.
Concrete = (0.18 − 0.125)π = 0.055π ≈ 0.1728 m³ (4 d.p.).
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The two rectangles are joined along a 4 cm edge, not overlapping. There is no shared area, so nothing should be subtracted — the cross-section is simply 24 + 12 = 36 cm². The lesson's misconception card warns: "do not add overlapping parts twice" — here there is no overlap.
(c) Corrected working:
Rectangle 1: 6 × 4 = 24 cm² ✓
Rectangle 2: 4 × 3 = 12 cm² ✓
Cross-section = 24 + 12 = 36 cm² (no overlap)
V = 36 × 10 = 360 cm³.
3 — Open-ended challenge (sample solution)
Sample problem: "A camping tent is a triangular prism. The two ends are isosceles triangles with base 12 m and slant edges of 10 m each (the slant edges run from the corners of the base up to the apex). The tent is 8 m long. Find the volume of air inside the tent."
Sketch: a triangular cross-section with base 12 m, slant edges 10 m, perpendicular height = "?". Drop a perpendicular from the apex; the base splits into 6 m + 6 m.
Solution using Pythagoras:
Half-base = 6 m, slant = 10 m, so perpendicular height h satisfies h² + 6² = 10² ⇒ h² = 100 − 36 = 64 ⇒ h = 8 m.
Atriangle = ½ × 12 × 8 = 48 m².
Vtent = 48 × 8 = 384 m³.
Marking: 1 for a clear word-problem description, 1 for a labelled sketch with a "?", 1 for a correct Pythagoras step yielding an integer (e.g. 8 m here from 6-8-10), 1 for the correct final volume with units. Other valid problems accepted (e.g. cone where slant l = 13 and r = 5 ⇒ h = 12).