You are wrapping a rectangular gift box in paper. You need to estimate how much paper you need. What information would you use? What would you calculate? How is this different from finding the volume of the box?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Triangular prism — $A_\triangle$ = area of triangular end, $P_\triangle$ = perimeter of triangle, $L$ = length of prismKey insight: lateral SA = perimeter of cross-section × length (works for any right prism)
Curved surface area of a cylinder = $2\pi r \times h$
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Think: rectangle with width = circumference $2\pi r$, height = $h$
Know
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
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
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
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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)
Misconceptions to Fix
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Wrong: Converting units only requires multiplying by 10.
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Right: Metric conversions use powers of 10, but area conversions use powers of 100 and volume uses powers of 1000.
Core Content
01
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?
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.
02
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
Three pairs of rectangles: top/bottom, front/back, left/right
Two identical triangular ends + three rectangles (one per side of the triangle)
Two circles (top and bottom) + one rectangle (curved surface unrolled)
Number of faces
6
5
3 components
The Cylinder Key Insight
When you unroll the curved surface of a cylinder, you get a rectangle. The width of that rectangle wraps once around the circle — so its width must equal the circumference:
$$\text{Curved SA} = 2\pi r \times h$$
This is not a separate formula to memorise — it is simply: rectangle area = width × height, where width = circumference.
03
Surface Area of a Rectangular Prism
Six faces in three pairs. Identify the pairs, calculate each pair, add.
$$\text{SA} = 2\ell w + 2\ell h + 2wh$$
Dimensions
$\ell \times w$
$\ell \times h$
$w \times h$
Area (each face)
$\ell w$
$\ell h$
$wh$
Three pairs of parallel faces — each pair has equal area
List every face before calculating. Write "Top = ..., Bottom = ..., Front = ..., Back = ..., Left = ..., Right = ..." and tick each one. The most common multi-mark error in this topic is missing one or two faces because you calculated quickly without tracking.
04
Surface Area of a Triangular Prism
Five faces: two triangular ends plus three rectangular side faces — one per edge of the triangle.
$$\text{SA} = 2 \times A_\triangle + (a + b + c) \times L$$
where $a$, $b$, $c$ are the side lengths of the triangular cross-section and $L$ is the length of the prism.
Pattern: $(a + b + c)$ is the perimeter of the triangular cross-section. Lateral SA = perimeter of cross-section × length. This works for any right prism — a useful shortcut for the three rectangular faces.
Right-angled triangle cross-section: If the triangle is right-angled, you need the hypotenuse for the third rectangular face. Use Pythagoras to find it before attempting the surface area. This is a deliberate two-step HSC question design.
05
Surface Area of a Cylinder
$$\text{SA} = 2\pi r^2 + 2\pi rh$$
Faces present
Two circles + curved surface
One circle + curved surface
Curved surface only
Formula
$2\pi r^2 + 2\pi rh$
$\pi r^2 + 2\pi rh$
$2\pi rh$
Curved surface unrolls to a rectangle: width = circumference (2πr), height = h
Diameter trap: If the diameter is given, write $r = d \div 2$ as the first line of working — every time, without exception. Substituting diameter directly into $\pi r^2$ gives an area four times too large.
Keep $\pi$ exact until the end: Compute $2\pi r^2 + 2\pi rh = (2r^2 + 2rh)\pi$. Factor out $\pi$, evaluate the bracket, multiply by $\pi$ at the very last step. Fewer rounding errors, cleaner working, full marks.
06
Common Mistakes
Mistake 1 — 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.
Mistake 2 — 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.
Mistake 3 — 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 units of your answer — if they are cubic when the question asks for surface area, you used the wrong formula.
Worked Examples
07
Worked Example 1
Surface Area — Rectangular Prism
Problem
Find the surface area of a rectangular prism with length 8 cm, width 5 cm, and height 3 cm.
Add the $\pi$ terms: $98 + 168 = 266$. Evaluate $266\pi$ at the very last step. Leaving intermediate answers as multiples of $\pi$ is expected in strong responses.
10
Worked Example 4
Surface Area — Open Cylinder (Practical)
Problem
A cylindrical water tank is open at the top. It has a diameter of 3 m and a height of 4 m. Find the surface area of material needed to construct the tank, correct to 2 decimal places.
Step-by-Step Solution
1
Identify version and find radius Open top → $\text{SA} = \pi r^2 + 2\pi rh$ $r = 3 \div 2 = 1.5$ m
Open at top = one circle (base only) + curved surface. Write $r = d \div 2$ before anything else.
Using the full formula (with top) would give $14.25\pi + 2.25\pi = 16.5\pi = 51.84$ m² — including 7.07 m² of material that the open-top tank does not need. Reading "open at the top" carefully differentiates marks.
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Surface Area of Prisms and Cylinders
List all faces before calculating. State which version of the cylinder formula you are using.
Section A — Rectangular Prisms
1 Find the surface area of a rectangular prism with $\ell = 10$ cm, $w = 4$ cm, $h = 6$ cm.
Working space in book
Saved
2 Find the surface area of a cube with side length 5 m.
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3 A rectangular box has no lid. Its base is 12 cm × 8 cm and its height is 5 cm. Find the surface area of material needed.
Working space in book
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Section B — Triangular Prisms
4 A triangular prism has an equilateral triangular cross-section with side 6 cm and prism length 10 cm. Find the total surface area. (Use sine area rule for triangle area, included angle 60°.)
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5 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 surface area.
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Section C — Cylinders
6 Find the total SA of a closed cylinder with $r = 4$ cm and $h = 9$ cm. Answer to 2 decimal places.
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7 Find the curved surface area only of a cylinder with $r = 3$ m and $h = 7$ m. Answer to 2 decimal places.
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8 A cylindrical tin can has diameter 10 cm and height 14 cm, with top and bottom lids. Find total SA to 2 decimal places.
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9 A section of pipe has inner radius 4 cm, outer radius 5 cm, and length 20 cm. Find the total SA (inner curved surface, outer curved surface, and two annular end faces) to 2 decimal places.
2 A rectangular prism has $\ell = 6$ m, $w = 4$ m, $h = 3$ m. One face measuring $6 \times 4$ m is open (no lid). Its surface area is:
A 72 m²
B 84 m²
C 108 m²
D 144 m²
? Regarding this topic, 2 A rectangular prism has $\ell = 6$ m, $w = 4$ m, $h = 3$ m. One face measuring $6 \times 4$ m is open (no lid). Its surface area is:
A 72 m²
B 84 m²
C 108 m²
D 144 m²
B - Correct!
B — Full SA $= 2(24) + 2(18) + 2(12) = 48 + 36 + 24 = 108$ m². Subtract open face $6 \times 4 = 24$: SA $= 108 - 24 = 84$ m².
3 A triangular prism has a right-angled cross-section with legs 3 cm and 4 cm (hypotenuse 5 cm). The prism is 10 cm long. Its total SA is:
A 96 cm²
B 108 cm²
C 132 cm²
D 144 cm²
? Regarding this topic, 3 A triangular prism has a right-angled cross-section with legs 3 cm and 4 cm (hypotenuse 5 cm). The prism is 10 cm long. Its total SA is:
A 96 cm²
B 108 cm²
C 132 cm²
D 144 cm²
C - Correct!
C — Two triangles: $2 \times \frac{1}{2}(3)(4) = 12$ cm². Three rectangles: $(3+4+5) \times 10 = 120$ cm². Total $= 132$ cm².
Short Answer
11
SA 42 marks
Find the total SA of a closed cylinder with diameter 10 cm and height 6 cm. Give your answer in terms of $\pi$.
Syllabus: MS11-3 | 1 mark correct $r$ + formula; 1 mark correct exact answer
Working space in book
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12
SA 53 marks
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.
Syllabus: MS11-3 | 1 mark hypotenuse; 1 mark all faces listed; 1 mark correct total
Working space in book
Saved
13
SA 64 marks
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)
(c) Find the total SA of the material used. (The top is open.) (2 marks)