Lesson 9 ~30 min Unit 3 · Measurement & Geometry +85 XP

Surface Area of Prisms & Cylinders

Unfold every 3D shape into a flat net — add the areas of all faces to find total surface area.

Today's hook: A factory wraps cylindrical tin cans — radius 4 cm, height 12 cm. How much tin sheet is needed for each can?

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Think First

warm-up

If you unwrapped a cardboard cereal box and laid it flat, how many rectangles would you see? How would you work out the total area of card used?

Record your answer in your workbook.

The Big Idea

+5 XP

Surface area (SA) is the total area of every face on a 3D shape. The best strategy is to unfold the shape into its net — a flat diagram showing every face — then add up all the areas. For a rectangular prism: $\text{SA} = 2(lw + lh + wh)$. For a cylinder: $\text{SA} = 2\pi r^2 + 2\pi r h$.

$$\text{SA}_\text{box} = 2(lw + lh + wh)$$

Net strategy

Always draw or imagine the net first. Count every face. Opposite faces on a rectangular prism are identical — pair them and multiply by 2.

Units matter

Surface area is always in square units: cm², m², mm². A 2D measurement of a 3D surface.

Cylinder faces

A cylinder has 2 circular ends + 1 curved surface. The curved surface unrolls into a rectangle with width $= 2\pi r$ and height $= h$.

What You'll Master

objectives

Know

SA = total area of all faces
Rectangular prism: $\text{SA} = 2(lw+lh+wh)$
Cylinder: $\text{SA} = 2\pi r^2 + 2\pi r h$

Understand

Why nets help — unfolding makes all faces visible
How the cylinder formula is derived (2 circles + rectangle)
Why there are pairs of equal faces on a rectangular prism

Can Do

Find SA of rectangular prisms from all three dimensions
Find SA of triangular prisms using base area + lateral faces
Find SA (or curved SA only) of cylinders
Solve real-world wrapping and packaging problems

Words You Need

vocabulary

Surface area (SA)Total area of every outer face of a 3D shape. Measured in square units.

NetA flat diagram formed by unfolding a 3D shape. Every face of the solid appears in the net.

PrismA 3D shape with two identical parallel cross-sections (bases) connected by rectangular lateral faces.

Lateral surfaceThe side faces of a prism — not the two bases. For a cylinder this is the curved surface.

CylinderA solid with two circular bases and a curved lateral surface. $\text{SA} = 2\pi r^2 + 2\pi r h$.

Triangular prismA prism whose cross-section is a triangle. SA = 2 × triangle base area + 3 rectangular lateral faces.

Spot the Trap

heads-up

Classic error: forgetting that a rectangular prism has three pairs of faces, not three faces.

Wrong (only 3 faces)

$\text{SA} = lw + lh + wh = 12 + 15 + 20 = 47$ cm² ✗

Correct (6 faces = 3 pairs)

$\text{SA} = 2(lw + lh + wh) = 2 \times 47 = 94$ cm² ✓

Rule: Always multiply the sum of the three face-type areas by 2.

Nets of Prisms

+5 XP

To find SA, unfold the prism into its net. A rectangular prism unfolds into a cross of 6 rectangles — 3 pairs of identical faces. A triangular prism unfolds into 2 triangles and 3 rectangles (whose widths equal the three sides of the triangle).

Count every face. Opposite faces on a rectangular prism are equal: top = bottom, front = back, left = right.

$$\text{SA}_\text{tri prism} = 2 \times A_\triangle + \text{sum of 3 rectangles}$$

SA of Rectangular Prisms

+5 XP

A rectangular prism (cuboid) has 6 rectangular faces in 3 pairs:

Top + Bottom: each $= l \times w$
Front + Back: each $= l \times h$
Left + Right: each $= w \times h$

$\text{SA} = 2(lw + lh + wh)$

$$\text{SA} = 2(lw + lh + wh)$$

Brackets first

Compute $lw + lh + wh$ inside the brackets first, then multiply the total by 2.

SA of Triangular Prisms

+5 XP

A triangular prism has 5 faces: 2 triangular ends + 3 rectangular lateral faces. The widths of the rectangles equal the three sides of the triangle ($a$, $b$, $c$) and all have the same height $l$ (length of prism).

$\text{SA} = 2 \times A_\triangle + (a + b + c) \times l$

For a right-angled triangle: $A_\triangle = \frac{1}{2} \times \text{base} \times \text{height}$.

$$\text{SA} = 2 \times A_\triangle + (a+b+c) \times l$$

SA of Cylinders

+5 XP

A cylinder has 3 parts: 2 circular ends + 1 curved surface. Unroll the curved surface — it becomes a rectangle with width $= 2\pi r$ (circumference) and height $= h$.

$\text{SA} = 2\pi r^2 + 2\pi r h$

Sometimes only the curved surface area (CSA) is needed: $\text{CSA} = 2\pi r h$.

$$\text{SA} = 2\pi r^2 + 2\pi r h$$

Factor out $2\pi r$

$\text{SA} = 2\pi r(r + h)$. Useful for quick calculation on a calculator.

Open vs closed cylinders

Open tin (no lid): SA $= \pi r^2 + 2\pi r h$ (only 1 circle). Full sealed can: $\text{SA} = 2\pi r^2 + 2\pi r h$.

Watch Me Solve It · 3 examples

WE 1 — SA of a Rectangular Box

+10 XP

PROBLEM

Find the SA of a rectangular box with $l = 4$ cm, $w = 3$ cm, $h = 5$ cm.

1
Write the formula
$$\text{SA} = 2(lw + lh + wh)$$
2
Calculate each pair
$lw = 4 \times 3 = 12$ $lh = 4 \times 5 = 20$ $wh = 3 \times 5 = 15$
3
Sum the three pairs
$12 + 20 + 15 = 47$
4
Multiply by 2
$\text{SA} = 2 \times 47 = 94$ cm²

Answer$\text{SA} = 94$ cm²

WE 2 — SA of a Triangular Prism

+10 XP

PROBLEM

An equilateral triangular prism has side length $s = 6$ cm and prism length $l = 10$ cm. Find SA (use $\sqrt{3} \approx 1.732$).

1
Find triangle area (equilateral)
$A_\triangle = \frac{\sqrt{3}}{4} s^2 = \frac{1.732}{4} \times 36 = 0.433 \times 36 \approx 15.59$ cm²
2
Find 3 rectangular lateral faces
$3 \times s \times l = 3 \times 6 \times 10 = 180$ cm²
3
Add 2 triangles + 3 rectangles
$\text{SA} = 2 \times 15.59 + 180 = 31.18 + 180 \approx 211$ cm²

Answer$\text{SA} \approx 211$ cm²

WE 3 — SA of a Cylinder

+10 XP

PROBLEM

Find the SA of a cylinder with $r = 3$ cm, $h = 8$ cm. Leave in exact form ($n\pi$) and as a decimal.

1
Write the formula
$$\text{SA} = 2\pi r^2 + 2\pi r h$$
2
Circles: $2\pi r^2$
$2 \times \pi \times 9 = 18\pi$
3
Curved surface: $2\pi r h$
$2 \times \pi \times 3 \times 8 = 48\pi$
4
Add and convert
$\text{SA} = 18\pi + 48\pi = 66\pi \approx 207.3$ cm²

Answer$\text{SA} = 66\pi \approx 207.3$ cm²

Common Pitfalls

heads-up

Forgetting to Double for Rectangular Prisms

Mistake: $\text{SA} = lw + lh + wh$ (only three faces counted).

Fix: $\text{SA} = 2(lw + lh + wh)$ — there are 3 pairs, not 3 faces.

Using Diameter Instead of Radius in Cylinder

Mistake: Plugging $d$ where $r$ belongs in $\text{SA} = 2\pi r^2 + 2\pi r h$.

Fix: Always use radius. $r = d/2$ before substituting.

Counting Faces Incorrectly for Triangular Prism

Mistake: Counting 2 rectangles instead of 3 (forgetting the base rectangle).

Fix: Draw the net. 2 triangles + 3 rectangles (one for each triangle side).

Copy Into Your Books

Rectangular prism: $\text{SA} = 2(lw + lh + wh)$ — 6 faces, 3 pairs

Triangular prism: $\text{SA} = 2A_\triangle + (a+b+c) \times l$

Cylinder (full): $\text{SA} = 2\pi r^2 + 2\pi rh$ CSA only: $2\pi rh$

Strategy: Draw the net first — count every face before calculating.

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Surface Area

4 problems

Set a timer for 4 minutes. Show all working.

1 Find SA of a cube with side 5 cm.

$\text{SA} = 6 \times 5^2 = 6 \times 25 = 150$ cm²
2 Find SA of box: $l = 6$, $w = 4$, $h = 2$ cm.

$2(24 + 12 + 8) = 2 \times 44 = 88$ cm²
3 Cylinder $r = 5$ cm, $h = 10$ cm. Find SA.

$2\pi(25) + 2\pi(5)(10) = 50\pi + 100\pi = 150\pi \approx 471$ cm²
4 Hook: cylindrical can $r = 4$ cm, $h = 12$ cm. Find full SA of tin sheet needed.

$2\pi(16) + 2\pi(4)(12) = 32\pi + 96\pi = 128\pi \approx 402$ cm²

Complete in your workbook.

Quick Check · 5 questions

How many faces does a triangular prism have?

+10 XP

SA of a box: $l = 4$ cm, $w = 4$ cm, $h = 2$ cm.

+10 XP

SA of a box: $l = 4$ cm, $w = 3$ cm, $h = 5$ cm.

+10 XP

SA of a cylinder: $r = 5$ cm, $h = 10$ cm.

+10 XP

SA of a cylinder: $r = 5$ cm, $h = 15$ cm.

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. A rectangular box has dimensions 8 cm × 5 cm × 3 cm. Find its surface area.

Show full working in your book.

ApplyMedium3 MARKS

Q7. A cylinder has radius 4 cm and height 15 cm. Find the curved surface area only. Leave your answer in terms of $\pi$ and as a decimal.

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ReasonHard3 MARKS

Q8. A triangular prism has an isosceles right-angled triangular cross-section with two legs of 6 cm each. The prism is 10 cm long. Find the total surface area. (Hypotenuse $= 6\sqrt{2} \approx 8.49$ cm.)

Show full working in your book.

Comprehensive Answers

MC: 1-C, 2-B, 3-A, 4-C, 5-A

Q6: $\text{SA} = 2(40 + 24 + 15) = 2 \times 79 = 158$ cm².

Q7: $\text{CSA} = 2\pi \times 4 \times 15 = 120\pi \approx 376.99$ cm².

Q8: $A_\triangle = 18$ cm². Lateral: $60 + 60 + 84.9 = 204.9$ cm². $\text{SA} = 36 + 204.9 \approx 241$ cm².

Stretch Challenge · +25 XP

Mystery Cuboid

A cuboid has total surface area 148 cm². Two of its dimensions are 6 cm and 4 cm. Find the third dimension.

Reveal solution

$148 = 2(6 \times 4 + 6h + 4h) = 2(24 + 10h)$. So $74 = 24 + 10h$, giving $10h = 50$, thus $h = 5$ cm.

Quick Review

Rectangular prism

$\text{SA} = 2(lw + lh + wh)$

Triangular prism

$\text{SA} = 2A_\triangle + (a+b+c)l$

Cylinder (full)

$\text{SA} = 2\pi r^2 + 2\pi rh$

Curved SA only

$\text{CSA} = 2\pi rh$

Net strategy

Unfold → identify all faces → add areas

Key pitfall

Rectangular prism: multiply sum by 2 (3 pairs)

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