Mathematics • Year 9 • Unit 4 • Lesson 5
Surface Area and Volume of Prisms and Cylinders
Build fluency with V = (cross-section area) × length for prisms, V = πr²h for cylinders, and SA = 2πr² + 2πrh. One step at a time, from a fully worked example through guided practice to independent problems.
1. I do, fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Find the volume and total surface area of a cylinder with radius 4 cm and height 10 cm. Leave answers in terms of π AND give a decimal to 1 dp.
Step 1, Write the formulas you need.
V = π r² h
SA = 2π r² + 2π r h
Reason: write them down BEFORE you substitute, that way you can't accidentally use the wrong one.
Step 2, Identify r and h.
r = 4 cm, h = 10 cm.
Reason: r is RADIUS, not diameter. Always check, many problems give the diameter to catch you out.
Step 3, Compute the VOLUME.
V = π × 4² × 10 = π × 16 × 10 = 160π cm³ ≈ 502.7 cm³
Reason: square the radius FIRST, then multiply by height, then by π.
Step 4, Compute the SURFACE AREA in two parts.
2π r² (two circular ends) = 2π × 4² = 32π
2π r h (curved/lateral surface) = 2π × 4 × 10 = 80π
Reason: the cylinder net has TWO circles + ONE rectangle. Adding them gives the total SA.
Step 5, Add and write the final answer with correct units.
SA = 32π + 80π = 112π cm² ≈ 351.9 cm²
Reason: SA is measured in cm² (area), V in cm³ (volume). Different shapes of units!
Answer: V = 160π cm³ ≈ 502.7 cm³; SA = 112π cm² ≈ 351.9 cm².
2. We do, fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Find the volume and total surface area of a rectangular prism with length 8 cm, width 5 cm, height 3 cm.
Step 1, Write the formulas:
V = l × w × ____ SA = 2(lw + l____ + w____)
Step 2, Substitute l = 8, w = 5, h = 3:
Step 3, Compute V:
V = 8 × 5 × ____ = ____________ cm³
Step 4, Compute each face product (then add them inside the brackets):
l w = 8 × 5 = ____
l h = 8 × ____ = ____
w h = 5 × ____ = ____
Sum inside the brackets = ____ + ____ + ____ = ____
Step 5, Multiply by 2 and state units:
SA = 2 × ____ = ____________ cm²
3. You do, independent practice
Show your working in the space under each problem. The first four are foundation (one formula). The middle two are standard (both formulas, or with a twist). The last two are extension (work backwards / reasoning).
Foundation, single formula
3.1 Find the volume of a cube with side 5 cm. 1 mark
3.2 Find the surface area of a cube with side 5 cm. 1 mark
3.3 Find the volume of a cylinder with radius 3 cm and height 7 cm. Leave answer in terms of π. 1 mark
3.4 Find the LATERAL (curved) surface area only of a cylinder with radius 2 cm and height 8 cm. Leave answer in terms of π. 1 mark
Standard, both formulas
3.5 Find the volume AND the total surface area of a rectangular prism with dimensions 6 × 4 × 2 cm. 2 marks
3.6 A cylinder has DIAMETER 10 cm and height 12 cm. Find its volume and total surface area to 1 dp. (Careful: diameter, not radius!) 2 marks
Extension, push your thinking
3.7 A cylinder has volume 100π cm³ and radius 5 cm. Find its height. 2 marks
3.8 A triangular prism has a right-triangle cross-section with legs 3 cm and 4 cm, and the prism is 10 cm long. Find its volume. (Hint: area of the right triangle = ½ × base × height.) 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (rectangular prism 8 × 5 × 3)
Step 1: V = l × w × h; SA = 2(lw + l h + w h).
Step 3: V = 8 × 5 × 3 = 120 cm³.
Step 4: lw = 40, lh = 8 × 3 = 24, wh = 5 × 3 = 15. Sum = 40 + 24 + 15 = 79.
Step 5: SA = 2 × 79 = 158 cm².
3.1, Cube volume, side 5
V = 5 × 5 × 5 = 5³ = 125 cm³.
3.2, Cube surface area, side 5
A cube has 6 equal square faces. SA = 6 × 5² = 6 × 25 = 150 cm².
3.3, Cylinder volume, r = 3, h = 7
V = π × 3² × 7 = 9 × 7 × π = 63π cm³ (≈ 197.9 cm³).
3.4, Cylinder lateral SA, r = 2, h = 8
Lateral SA = 2π r h = 2π × 2 × 8 = 32π cm² (≈ 100.5 cm²). This is only the curved part, does NOT include the two circular ends.
3.5, Prism 6 × 4 × 2
V = 6 × 4 × 2 = 48 cm³.
SA = 2(6×4 + 6×2 + 4×2) = 2(24 + 12 + 8) = 2(44) = 88 cm².
3.6, Cylinder DIAMETER 10, height 12
Diameter = 10 → radius = 5 cm. (Halve the diameter first.)
V = π × 5² × 12 = π × 25 × 12 = 300π ≈ 942.5 cm³.
SA = 2π × 5² + 2π × 5 × 12 = 50π + 120π = 170π ≈ 534.1 cm².
If you forgot to halve and used r = 10, you'd get 4× the correct volume, a common error.
3.7, Cylinder V = 100π, r = 5
V = π r² h → 100π = π × 25 × h → 100 = 25h → h = 4 cm.
3.8, Triangular prism, legs 3 cm and 4 cm, length 10 cm
Cross-section area = ½ × 3 × 4 = 6 cm².
V = 6 × 10 = 60 cm³.
This works for ANY prism: V = (cross-section area) × length.