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Module 2 · L7 of 22 ~55 min MS-M1 · MED-HIGH ⚡ +95 XP available

Surface Area of Prisms and Cylinders

Unfold the solid into a net. Every face appears exactly once. Add them all, then subtract any faces that are missing.

Today's hook, You are wrapping a rectangular gift box in paper. What information do you need to know how much paper to buy, and how is this different from finding the volume of the box?

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Recall, your gut answer first

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Worksheets

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Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question

Recall, your gut answer first

+5 XP warm-up

You are wrapping a rectangular gift box in paper. You need to estimate how much paper you need.

Without any formula what information would you use? What would you calculate? How is this different from finding the volume of the box?

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Surface area formulas you need to own

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Three solids, three formulas. One key insight links them all: surface area = sum of all face areas. A net diagram makes every face visible so you cannot accidentally omit one.

Rectangular prism: Three pairs of rectangles, top/bottom, front/back, left/right. Triangular prism: Two triangular ends plus three rectangular side faces (one per triangle edge). The lateral SA = perimeter of cross-section × length for any right prism.

Surface-area formulas for a rectangular prism and a cylinder.

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

Rectangular prism, 6 faces

$\text{SA} = 2\ell w + 2\ell h + 2wh$. Three pairs, list each pair, calculate, add.

Triangular prism, 5 faces

$\text{SA} = 2A_\triangle + (a+b+c) \times L$. Lateral SA = perimeter of cross-section × length.

Cylinder, closed or open

Closed: $2\pi r^2 + 2\pi rh$. Open top: $\pi r^2 + 2\pi rh$. Pipe: $2\pi rh$ only.

What you'll master

Know

Key facts

What surface area means and how net diagrams represent it
The SA formula for a cylinder: $\text{SA} = 2\pi r^2 + 2\pi rh$
How to handle open or partial surface area problems

Understand

Concepts

Why surface area = sum of all face areas, and why a net makes every face visible
Why the cylinder's curved surface "unrolls" into a rectangle of width $2\pi r$
Why removing a face means subtracting its area from the total

Can do

Skills

Draw or describe the net of any prism or cylinder and use it to find total SA
Calculate SA of any right prism or cylinder
Adjust calculations for open-top containers, pipes, and partial solids

Key vocabulary

Surface areaThe total area of all outer faces of a 3D solid, measured in square units.

NetA 2D diagram showing all faces of a solid "unfolded" flat, every face appears exactly once.

Right prismA solid with two identical parallel bases connected by rectangular faces perpendicular to the base.

Cross-sectionThe shape when you slice through a prism parallel to its base, identical at every cut.

Curved surface areaArea of the curved face of a cylinder, found by "unrolling" it into a rectangle.

Open solidA solid missing one or more faces, common in container problems (box with no lid, pipe with no ends).

What surface area means

Surface area answers: if you peeled off every face of this solid and laid them flat, what would the total area be?

What surface area means

core concept

Surface area answers: if you peeled off every face of this solid and laid them flat, what would the total area be?

This is directly useful: calculating how much material to build a container, how much paint to cover a surface, how much foil to wrap a package.

The Method, Always Three Steps

Identify every face of the solid
Find the area of each face
Add them all together

The net diagram makes step 1 reliable, when you unfold the solid, every face is visible and you cannot accidentally omit one.

Surface area vs volume: These are frequently confused. Surface area = outside skin, measured in square units (cm², m²). Volume = space inside, measured in cubic units (cm³, m³). Before writing any formula, confirm which one the question asks for. Writing "SA =" commits you to the correct formula.

Net Diagrams

A net is what you get when you cut along some edges of a solid and unfold it completely flat. Every face appears exactly once.

Net components

Rectangular prism: Three pairs of rectangles (top/bottom, front/back, left/right), 6 faces.

Triangular prism: Two identical triangular ends + three rectangles (one per side), 5 faces.

Cylinder: Two circles (top and bottom) + one rectangle (curved surface unrolled), 3 components.

The Cylinder Key Insight

When you unroll the curved surface of a cylinder, you get a rectangle. The width wraps once around the circle, so its width equals the circumference:

$\text{Curved SA} = 2\pi r \times h$

This is simply: rectangle area = width × height, where width = circumference.

Surface area = total area of all faces unfolded flat. For prisms: identify every face (top, bottom, and all lateral faces), calculate each separately, then sum. The net is a useful way to count faces systematically.

Pause, copy the surface area definition (total area of all faces unfolded flat), the net method (draw and label every face), and the formula for a cylinder's surface area (SA = 2πr² + 2πrh) into your book.

Did you get this? True or false: surface area is measured in cubic units (cm³) because solids are three-dimensional.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · RECTANGULAR PRISM SA

Find the surface area of a rectangular prism with length 8 cm, width 5 cm, and height 3 cm.

Identify all faces: $\ell = 8$, $w = 5$, $h = 3$
3 pairs: top/bottom, front/back, left/right

List the three pairs before calculating, ensures no face is missed.

PROBLEM 2 · TRIANGULAR PRISM SA

A triangular prism has a right-angled triangular cross-section with legs 6 cm and 8 cm. The prism is 15 cm long. Find the total surface area.

Find the hypotenuse first (Pythagoras)
$c^2 = 6^2 + 8^2 = 36 + 64 = 100$
$c = 10\text{ cm}$

Need the hypotenuse for the third rectangular face. This Pythagoras step must come before the SA formula.

PROBLEM 3 · CYLINDER SA (CLOSED AND OPEN)

(a) Find the total SA of a closed cylinder with radius 7 cm and height 12 cm, correct to 2 decimal places.
(b) A cylindrical water tank is open at the top with diameter 3 m and height 4 m. Find the material needed, correct to 2 decimal places.

a–1

Closed cylinder, $r = 7$, $h = 12$
$\text{SA} = 2\pi r^2 + 2\pi rh$
$= 2\pi(49) + 2\pi(7)(12) = 98\pi + 168\pi = 266\pi$

Both ends present, use the full formula. Keep $\pi$ exact: $2 \times 49 = 98$, $2 \times 84 = 168$.

Quick check: A closed cylinder has radius 3 cm and height 8 cm. Its total SA is:

Common errors · the 3 traps that cost marks

Trap 01

Missing one or more faces

Calculate four faces of a rectangular prism instead of six, or four of a triangular prism instead of five. Fix: list every face with its dimensions before calculating. Tick each one as you go. 30 extra seconds prevents multi-mark errors.

Trap 02

Diameter instead of radius in the cylinder formula

Substituting diameter $d$ into SA $= 2\pi r^2 + 2\pi rh$ inflates each term by a factor of 4 (for the circle) or 2 (for the curved surface). Write $r = d \div 2$ first. Always.

Trap 03

Surface area vs volume confusion

Both involve the same dimensions. Surface area → square units (cm², m²), outside skin. Volume → cubic units (cm³, m³), space inside. Check the units in your answer, if they are cubic when the question asks for surface area, you used the wrong formula.

Fill the gap: A closed cylinder has radius 4 cm and height 9 cm. $\text{SA} = 2\pi($$) + 2\pi(4)(9) = 32\pi +$ $\pi = 104\pi \approx 326.73\text{ cm}^2$.

Quick-fire practice · 9 calculations

Find the SA of a rectangular prism with $\ell = 10$ cm, $w = 4$ cm, $h = 6$ cm.

Find the SA of a cube with side length 5 m.

A rectangular box has no lid. Its base is 12 cm × 8 cm and height is 5 cm. Find the SA of material needed.

A triangular prism has an equilateral triangular cross-section with side 6 cm and prism length 10 cm. Find the total SA. (Use $A = \frac{1}{2} \times 6 \times 6 \times \sin 60°$ for triangle area.)

A triangular prism has a right-angled triangular cross-section with legs 5 m and 12 m, and prism length 8 m. Find the total SA.

Find the total SA of a closed cylinder with $r = 4$ cm and $h = 9$ cm. Answer to 2 decimal places.

Find the curved surface area only of a cylinder with $r = 3$ m and $h = 7$ m. Answer to 2 decimal places.

A cylindrical tin can has diameter 10 cm and height 14 cm, with top and bottom lids. Find total SA to 2 decimal places.

A section of pipe has inner radius 4 cm, outer radius 5 cm, and length 20 cm. Find total SA (inner curved, outer curved, two annular ends) to 2 decimal places.

Match each solid to its surface area formula:

Closed cylinder

$2\pi r^2 + 2\pi rh$

Revisit your thinking

Earlier you described what you would need to wrap a rectangular gift box. Look back at your initial response.

You need the three dimensions ($\ell$, $w$, $h$), and you calculate the total area of all six faces: $\text{SA} = 2\ell w + 2\ell h + 2wh$. This is different from volume ($\ell \times w \times h$) which measures the space inside in cubic units, surface area measures the outside skin in square units.

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Odd one out: Three of these statements about surface area are correct. Which one is wrong?

Show what you have learned

Multiple choice, then short answer under exam conditions.

Multiple choice

+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence. That tells the system what to drill next. Each retry pulls a fresh mix from the bank.

Short answer

ApplyBand 32 marks

SA 4. Find the total SA of a closed cylinder with diameter 10 cm and height 6 cm. Give your answer in terms of $\pi$. (2 marks)

1 mark correct $r$ + formula; 1 mark correct exact answer

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ApplyBand 43 marks

SA 5. A chocolate box is a triangular prism. The triangular ends are right-angled triangles with legs 9 cm and 12 cm. The box is 20 cm long. Find the total SA. (3 marks)

1 mark hypotenuse; 1 mark all faces listed; 1 mark correct total

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AnalyseBand 54 marks

SA 6. A water trough is a triangular prism lying on its side. The cross-section is an isosceles triangle with two equal sides of 50 cm and a base of 60 cm. The trough is 120 cm long and open at the top.

(a) Find the perpendicular height of the triangular cross-section. (1 mark)

(b) Find the area of one triangular end face. (1 mark)

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Comprehensive answers (click to reveal)

Drill 1: $2(40) + 2(60) + 2(24) = 80 + 120 + 48 = \mathbf{248\text{ cm}^2}$

Drill 2: $\text{SA} = 6 \times 25 = \mathbf{150\text{ m}^2}$

Drill 3: Full: $2(96) + 2(60) + 2(40) = 392$ cm²; subtract lid $96$: $\mathbf{296\text{ cm}^2}$

Drill 4: $A_\triangle = \frac{1}{2}(36)\sin 60° = 15.59$ cm²; $2 \times 15.59 + 3 \times (6 \times 10) = 31.18 + 180 = \mathbf{211.18\text{ cm}^2}$

Drill 5: Hyp $= 13$ m; $2 \times (\frac{1}{2}(5)(12)) = 60$ m²; Rects: $5(8)+12(8)+13(8) = 240$; Total $= \mathbf{300\text{ m}^2}$

Drill 6: $2\pi(16) + 2\pi(36) = 32\pi + 72\pi = 104\pi = \mathbf{326.73\text{ cm}^2}$

Drill 7: $2\pi(3)(7) = 42\pi = \mathbf{131.95\text{ m}^2}$

Drill 8: $r=5$; $2\pi(25) + 2\pi(5)(14) = 50\pi + 140\pi = 190\pi = \mathbf{596.90\text{ cm}^2}$

Drill 9: Outer: $200\pi$; Inner: $160\pi$; Annuli: $18\pi$; Total $= 378\pi = \mathbf{1187.52\text{ cm}^2}$

SA 4: $r = 5$; $\text{SA} = 2\pi(25) + 2\pi(5)(6) = 50\pi + 60\pi = \mathbf{110\pi\text{ cm}^2}$

SA 5: $c = 15$ cm; Triangles: $2 \times 54 = 108$ cm²; Rects: $9(20)+12(20)+15(20) = 720$; Total $= \mathbf{828\text{ cm}^2}$

SA 6(a): $h^2 = 50^2 - 30^2 = 1600$; $h = \mathbf{40\text{ cm}}$

SA 6(b): $A = \frac{1}{2}(60)(40) = \mathbf{1200\text{ cm}^2}$

SA 6(c): 2 triangles: 2400 cm²; 2 sloped faces: $2 \times 50 \times 120 = 12000$ cm²; base: $60 \times 120 = 7200$ cm²; Total $= \mathbf{21600\text{ cm}^2}$

Consolidate and move on

Sit the module quiz, then close the lesson off.

Boss battle · Surface Area Sprint

earn bronze · silver · gold

Five timed questions on surface area of prisms and cylinders. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Pool: lessons 1–7. Replays welcome.

⚔ Enter the arena

Mark lesson as complete

Tick when you've finished the practice and review.

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Module 2 overview · Year 11 Maths Standard