← Unit 4 Surface Area and Volume of Prisms and Cylinders

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Surface Area and Volume of Prisms and Cylinders

⏱ 25 min📚 Year 9📈
Think First

A cube has side 4 cm. What is its total surface area? What about a cylinder with radius 3 cm and height 5 cm?

💡 Revision: Ensure you know area formulas for rectangles, triangles, and circles.

Learning Intentions

Know

  • Prism cross-section
  • Lateral surface area
  • Total surface area of cylinder

Understand

  • Why a prism's volume depends on its cross-section
  • How unfolding a cylinder reveals its net

Can Do

  • Find surface area of rectangular prisms
  • Find surface area of cylinders
  • Decompose composite solids
Cross-sectionLateral areaNetSurface area
Learn Phase
1

Prisms

Constant cross-section

A prism is a solid with a constant cross-section. Its volume is:

Volume of Prism

$V = ext{Area of cross-section} imes ext{height (or length)}$

The surface area is the sum of all face areas.

Rectangular prism: $SA = 2(lw + lh + wh)$

2

Cylinders

A prism with circular cross-section

A cylinder has two circular bases and a curved side.

Cylinder Formulas

$V = pi r^2 h$

$SA = 2pi r^2 + 2pi r h$

The term $2pi r h$ is the lateral (curved) surface area.

Imagine unrolling the cylinder: the curved surface becomes a rectangle with dimensions $2pi r imes h$.

3

Nets

Unfolding 3D shapes

A net is a 2D pattern that folds into a 3D shape. Drawing nets helps visualise surface area.

For a cylinder, the net consists of:

  • Two circles (top and bottom)
  • One rectangle (curved side)

The rectangle's width equals the circle's circumference: $2pi r$.

✓

Check Understanding

Try it yourself

Find the total surface area of a cylinder with radius 4 cm and height 10 cm.

Worked Example

Surface Area and Volume of Prisms and Cylinders

1

Find the surface area of a rectangular prism 5 cm × 4 cm × 3 cm.

$SA = 2(5 imes4 + 5 imes3 + 4 imes3)$

$= 2(20 + 15 + 12) = 2(47) = 94$ cm²

2

A cylinder has radius 3 cm and height 8 cm. Find its surface area.

$SA = 2pi(3)^2 + 2pi(3)(8)$

$= 18pi + 48pi = 66pi approx 207.3$ cm²

3

A triangular prism has cross-section area 12 cm², length 10 cm. Find volume.

$V = 12 imes 10 = 120$ cm³

Common Misconceptions

✕

Use diameter instead of radius in cylinder formulas. All cylinder formulas use radius. If given diameter, halve it first.

✕

Confuse surface area with volume. Surface area is the sum of face areas (units are cm²). Volume is the space inside (units are cm³).

✕

Forget both circular ends of a cylinder. Total SA includes two circles: $2pi r^2$.

Your Turn

Practice — Prisms and Cylinders

Work through each question in your book or digitally. Answers are in the Questions phase.

1Find the surface area of a cube with side 6 cm.
2A cylinder has radius 5 cm, height 12 cm. Find its volume and surface area.
3A prism has a triangular cross-section (base 4 cm, height 3 cm) and length 10 cm. Find volume.
Real-World Anchor

Manufacturing and Packaging

Companies calculate surface area to determine material costs for boxes, cans, and containers. Minimising surface area reduces packaging costs and environmental impact. Australian recycling targets depend on efficient material use.

📓 Copy Into Your Books

▼

Rectangular prism

  • $SA = 2(lw + lh + wh)$
  • $V = lwh$

Cylinder

  • $V = pi r^2 h$
  • $SA = 2pi r^2 + 2pi r h$

General prism

  • $V = ext{cross-section area} imes ext{length}$
  • $SA = ext{sum of all faces}$
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
Surface area of cube side 3 cm:
Cylinder radius 2 cm, height 5 cm. Lateral SA:
Volume of cylinder radius 3 cm, height 4 cm:
A prism with cross-section area 8 cm², length 5 cm has volume:
Surface area units are:
Cylinder net has:
Rectangular prism 4×3×2 cm has SA:
The curved surface of a cylinder unrolls to a rectangle with width:
1Find the surface area of a cylinder with radius 6 cm and height 8 cm.
2A rectangular prism has dimensions 10 cm × 6 cm × 4 cm. Find its volume and surface area.
3Explain why the lateral surface area of a cylinder is $2pi r h$.

Comprehensive Answers

1Cylinder radius 6, height 8. SA?
$2pi(36) + 2pi(6)(8) = 72pi + 96pi = 168pi approx 527.8$ cm².
2Prism 10×6×4. V and SA?
$V = 240$ cm³. $SA = 2(60+40+24) = 248$ cm².
3Why lateral SA = $2pi r h$?
Unrolling gives rectangle: width = circumference $2pi r$, height = $h$. Area = $2pi r h$.
MC 1Cube side 3. SA?
$6 imes 9 = 54$ cm². Answer: C
MC 2Cylinder lateral SA.
$2pi(2)(5) = 20pi$. Answer: C
MC 3Cylinder volume.
$pi(3)^2(4) = 36pi$. Answer: C
MC 4Prism volume.
$8 imes 5 = 40$ cm³. Answer: B
MC 5SA units.
cm². Answer: B
MC 6Cylinder net.
2 circles + 1 rectangle. Answer: B
MC 7Prism 4×3×2 SA.
$2(12+8+6) = 52$ cm². Answer: B
MC 8Unrolled rectangle width.
Circumference $2pi r$. Answer: D
SA 1Cylinder SA radius 6, height 8.
$168pi approx 527.8$ cm².
SA 2Prism 10×6×4.
$V=240$ cm³, $SA=248$ cm².
SA 3Why $2pi r h$.
Unrolled circumference × height.
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