Singapore Math 5A · Independent learning

Volume of Complex Shapes

Learning target: Split or subtract nonoverlapping rectangular prisms to calculate composite volume.

Begin with the exact prerequisite checkpoint below. Afterward, explain why your strategy works, not merely which procedure to use.

Opening task

Explore · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 5 × 5 × 5 cm; B measures 4 × 2 × 2 cm. What is their combined volume?

Show answer and worked explanation

Answer: 141 cm³

How it works: Nonoverlapping A: 5×5×5=125 cm³; B: 4×2×2=16 cm³. Sum 125+16=141 cm³.

Lesson-specific prerequisite checkpoint

Do this without a worked example. If you cannot explain it, review the earlier skill before continuing.

Find 3 × 4 × 5.

Show answer and worked explanation

Answer: 60

How it works: Multiplying edge lengths yields volume in cubic units.

Understand the idea

Split or subtract nonoverlapping rectangular prisms to calculate composite volume.

Complex solids must be split into nonoverlapping cuboids, or computed as a full block minus a clearly specified cutout. Shared faces do not create duplicated volume, but overlapping interior regions must not be counted twice.

Vocabulary focus: unit cube, cuboid, cubic units, capacity. Make a small labeled model, work through the arithmetic, and explain the result in words.

Watch for: Double-counting shared cubes in overlapping prisms.

Model to study: Exploded/cutaway cuboid decomposition.

Worked examples

Worked examples 1 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 9 × 6 × 4 cm; B measures 3 × 2 × 2 cm. What is their combined volume?

Solution: 228 cm³

Why: Nonoverlapping A: 9×6×4=216 cm³; B: 3×2×2=12 cm³. Sum 216+12=228 cm³.

Worked examples 2 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 8 × 4 × 4 cm; B measures 3 × 2 × 4 cm. What is their combined volume?

Solution: 152 cm³

Why: Nonoverlapping A: 8×4×4=128 cm³; B: 3×2×4=24 cm³. Sum 128+24=152 cm³.

Worked examples 3 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 8 × 8 × 5 cm; B measures 4 × 2 × 5 cm. What is their combined volume?

Solution: 360 cm³

Why: Nonoverlapping A: 8×8×5=320 cm³; B: 4×2×5=40 cm³. Sum 320+40=360 cm³.

Check understanding

Check understanding 1 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 5 × 5 × 2 cm; B measures 2 × 4 × 5 cm. What is their combined volume?

Show answer and worked explanation

Answer: 90 cm³

How it works: Nonoverlapping A: 5×5×2=50 cm³; B: 2×4×5=40 cm³. Sum 50+40=90 cm³.

Check understanding 2 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 9 × 8 × 4 cm; B measures 3 × 2 × 5 cm. What is their combined volume?

Show answer and worked explanation

Answer: 318 cm³

How it works: Nonoverlapping A: 9×8×4=288 cm³; B: 3×2×5=30 cm³. Sum 288+30=318 cm³.

Guided practice

Guided practice 1 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 6 × 5 × 2 cm; B measures 3 × 2 × 3 cm. What is their combined volume?

Show answer and worked explanation

Answer: 78 cm³

How it works: Nonoverlapping A: 6×5×2=60 cm³; B: 3×2×3=18 cm³. Sum 60+18=78 cm³.

Guided practice 2 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 9 × 5 × 3 cm; B measures 3 × 4 × 2 cm. What is their combined volume?

Show answer and worked explanation

Answer: 159 cm³

How it works: Nonoverlapping A: 9×5×3=135 cm³; B: 3×4×2=24 cm³. Sum 135+24=159 cm³.

Guided practice 3 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 5 × 6 × 4 cm; B measures 4 × 2 × 2 cm. What is their combined volume?

Show answer and worked explanation

Answer: 136 cm³

How it works: Nonoverlapping A: 5×6×4=120 cm³; B: 4×2×2=16 cm³. Sum 120+16=136 cm³.

Independent practice

Independent practice 1 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 5 × 7 × 2 cm; B measures 4 × 4 × 4 cm. What is their combined volume?

Show answer and worked explanation

Answer: 134 cm³

How it works: Nonoverlapping A: 5×7×2=70 cm³; B: 4×4×4=64 cm³. Sum 70+64=134 cm³.

Independent practice 2 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 8 × 6 × 2 cm; B measures 3 × 3 × 4 cm. What is their combined volume?

Show answer and worked explanation

Answer: 132 cm³

How it works: Nonoverlapping A: 8×6×2=96 cm³; B: 3×3×4=36 cm³. Sum 96+36=132 cm³.

Independent practice 3 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 6 × 7 × 3 cm; B measures 4 × 2 × 2 cm. What is their combined volume?

Show answer and worked explanation

Answer: 142 cm³

How it works: Nonoverlapping A: 6×7×3=126 cm³; B: 4×2×2=16 cm³. Sum 126+16=142 cm³.

Independent practice 4 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 5 × 6 × 4 cm; B measures 3 × 3 × 5 cm. What is their combined volume?

Show answer and worked explanation

Answer: 165 cm³

How it works: Nonoverlapping A: 5×6×4=120 cm³; B: 3×3×5=45 cm³. Sum 120+45=165 cm³.

Independent practice 5 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 7 × 6 × 2 cm; B measures 4 × 3 × 5 cm. What is their combined volume?

Show answer and worked explanation

Answer: 144 cm³

How it works: Nonoverlapping A: 7×6×2=84 cm³; B: 4×3×5=60 cm³. Sum 84+60=144 cm³.

Challenge and error analysis

Challenge and error analysis 1 · worked-error diagnosis

Error-analysis notebook: Learner notebook (LESSON): “Combined volume = 175×32=5600 cm³; I multiplied disjoint cuboid volumes.” Identify exactly which written step or assumption fails, correct it, and independently solve this new task: Two cuboids are placed next to each other without overlap. Cuboid A measures 7 × 5 × 5 cm; B measures 2 × 4 × 4 cm. What is their combined volume?

Show answer and worked explanation

Answer: 207 cm³

How it works: Wrong step/assumption: Learner notebook (LESSON): “Combined volume = 175×32=5600 cm³; I multiplied disjoint cuboid volumes.” Diagnosis: Count full three-dimensional layers, use three perpendicular dimensions and cubic units; use cm³=mL only for capacity conversion. Correct computation: Nonoverlapping A: 7×5×5=175 cm³; B: 2×4×4=32 cm³. Sum 175+32=207 cm³.

Challenge and error analysis 2 · reasoning transfer

Reasoning and transfer: Solve, then explain a different way to check your answer (inverse operation, estimate, model, or units as appropriate). Two cuboids are placed next to each other without overlap. Cuboid A measures 6 × 6 × 2 cm; B measures 2 × 2 × 2 cm. What is their combined volume?

Show answer and worked explanation

Answer: 80 cm³

How it works: Nonoverlapping A: 6×6×2=72 cm³; B: 2×2×2=8 cm³. Sum 72+8=80 cm³. Verification: show why the answer has a reasonable magnitude and matches the operation, referent and unit.

Exit check

Exit check 1 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 8 × 5 × 5 cm; B measures 2 × 4 × 3 cm. What is their combined volume?

Show answer and worked explanation

Answer: 224 cm³

How it works: Nonoverlapping A: 8×5×5=200 cm³; B: 2×4×3=24 cm³. Sum 200+24=224 cm³.

Exit check 2 · composite volume

Two cuboids are placed next to each other without overlap. Cuboid A measures 7 × 5 × 3 cm; B measures 4 × 3 × 5 cm. What is their combined volume?

Show answer and worked explanation

Answer: 165 cm³

How it works: Nonoverlapping A: 7×5×3=105 cm³; B: 4×3×5=60 cm³. Sum 105+60=165 cm³.

Target for independent mastery: all exit-check problems correct plus a justified explanation. A tutor may reteach the concept if calculations are correct without meaningful reasoning.