Mathematics • Year 8 • Unit 3 • Lesson 9

Surface Area of Prisms & Cylinders

Build fluency with SA = 2(lw + lh + wh) for rectangular prisms and SA = 2πr² + 2πrh for cylinders. One worked example, one guided example, then eight graduated problems.

Build · I Do / We Do / You Do

1. I do — fully worked example

Watch the strategy: identify pairs of faces → multiply each → sum → multiply by 2.

Problem. Find the surface area of a rectangular box with l = 5 cm, w = 4 cm, h = 3 cm.

Step 1 — Write the formula.

SA = 2(lw + lh + wh)

Reason: a rectangular prism has 3 PAIRS of identical faces (top/bottom, front/back, left/right). The 2 multiplies in the pairs.

Step 2 — Calculate each face pair (inside the bracket first).

lw = 5 × 4 = 20 lh = 5 × 3 = 15 wh = 4 × 3 = 12

Step 3 — Sum the three pair-areas.

20 + 15 + 12 = 47

Step 4 — Multiply by 2 (for the pair partners).

SA = 2 × 47 = 94 cm²

Answer: SA = 94 cm² (always square units).

Stuck? Revisit lesson § Card 4 — "Spot the Trap": a box has SIX faces, not three. Always × 2.

2. We do — fill in the missing steps

A box with l = 6 cm, w = 4 cm, h = 2 cm. Fill in each blank. 4 marks

Step 1 — Write the formula:

SA = ______ × (lw + lh + wh)

Step 2 — Calculate each face pair:

lw = 6 × 4 = ______ lh = 6 × 2 = ______ wh = 4 × 2 = ______

Step 3 — Sum:

______ + ______ + ______ = ______

Step 4 — Multiply by 2:

SA = 2 × ______ = ______ cm²

Stuck? Revisit lesson § Card 6 — the brackets-first rule: sum the three face areas before multiplying by 2.

3. You do — independent practice

Show all working. Foundation: rectangular prisms with whole numbers. Standard: cylinders. Extension: real packaging.

Foundation — rectangular prisms

3.1 Find SA of a cube with side 5 cm. (Hint: a cube has 6 identical square faces, so SA = 6 × s².) 1 mark

3.2 Box: l = 4 cm, w = 4 cm, h = 2 cm. Find SA. 1 mark

3.3 Box: l = 10 cm, w = 6 cm, h = 4 cm. Find SA. 2 marks

3.4 Box: l = 12 cm, w = 8 cm, h = 5 cm. Find SA. 2 marks

Standard — cylinders

3.5 Cylinder: r = 5 cm, h = 10 cm. Find SA in exact form (nπ) and as a decimal (1 d.p.). 2 marks

3.6 Cylinder: r = 3 cm, h = 8 cm. Find SA in exact form (nπ) and as a decimal (1 d.p.). 2 marks

Extension — triangular prisms and curved-surface-only

3.7 A triangular prism has a right-angled triangular cross-section with legs 6 cm and 8 cm (hypotenuse = 10 cm) and prism length l = 12 cm. Find SA. (Hint: SA = 2 × (½ × 6 × 8) + (6 + 8 + 10) × 12.) 2 marks

3.8 An OPEN cylindrical tin (no lid) has r = 4 cm and h = 10 cm. Find the area of tin sheet needed — that's 1 base circle + the curved surface only. 2 marks

Stuck on 3.7 / 3.8? Triangular prism = 2 triangle ends + 3 rectangles (one per triangle side). Open cylinder = πr² + 2πrh (only ONE circle).

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Section 2 — We do (6 × 4 × 2 box)

Step 1: SA = 2 × (lw + lh + wh).
Step 2: lw = 24, lh = 12, wh = 8.
Step 3: 24 + 12 + 8 = 44.
Step 4: SA = 2 × 44 = 88 cm².

3.1 — Cube, side 5

SA = 6 × 5² = 6 × 25 = 150 cm².

3.2 — Box 4 × 4 × 2

lw = 16, lh = 8, wh = 8. Sum = 32. SA = 2 × 32 = 64 cm².

3.3 — Box 10 × 6 × 4

lw = 60, lh = 40, wh = 24. Sum = 124. SA = 2 × 124 = 248 cm².

3.4 — Box 12 × 8 × 5

lw = 96, lh = 60, wh = 40. Sum = 196. SA = 2 × 196 = 392 cm².

3.5 — Cylinder r = 5, h = 10

SA = 2π(5)² + 2π(5)(10) = 50π + 100π = 150π ≈ 471.2 cm².

3.6 — Cylinder r = 3, h = 8

SA = 2π(3)² + 2π(3)(8) = 18π + 48π = 66π ≈ 207.3 cm².

3.7 — Triangular prism (6-8-10 legs, l = 12)

Triangle area = ½ × 6 × 8 = 24 cm². Two triangles = 2 × 24 = 48 cm².
Three rectangles = (6 + 8 + 10) × 12 = 24 × 12 = 288 cm².
SA = 48 + 288 = 336 cm².

3.8 — Open cylinder r = 4, h = 10

SA = πr² + 2πrh = π(16) + 2π(4)(10) = 16π + 80π = 96π ≈ 301.6 cm². (Only ONE circle because there's no lid.)