Functional Skills Level 2 — Geometry and Measures

Surface Area of 3D Shapes

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Surface Area of a Cuboid

The surface area of a 3D shape is the total area of all its faces. For a cuboid, there are 3 pairs of identical rectangular faces.

\[SA_{\text{cuboid}} = 2(lw + lh + wh)\]
\(l\) = length, \(w\) = width, \(h\) = height
Net of a Cuboid — area of each face Top: l×w Left: w×h Front: l×h Right: w×h Back: l×h Bottom: l×w SA Formula ■ Top+Bottom = 2×l×w ■ Front+Back = 2×l×h ■ Left+Right = 2×w×h SA = 2(lw+lh+wh) Each colour = one pair of identical faces Deriving the Formula from the Net (example: 6×4×3 cm) Top + Bottom 2 × (6×4) = 2 × 24 = 48 cm² + Front + Back 2 × (6×3) = 2 × 18 = 36 cm² + Left + Right 2 × (4×3) = 2 × 12 = 24 cm² Total SA = 48 + 36 + 24 = 108 cm²

Example 1 — Find the surface area of a cuboid 8 cm × 5 cm × 3 cm

Step 1
Identify: \(l = 8\), \(w = 5\), \(h = 3\) (all in cm).
Step 2
Calculate each pair: \(lw = 8 \times 5 = 40\), \(lh = 8 \times 3 = 24\), \(wh = 5 \times 3 = 15\)
Step 3
Sum the pairs: \(40 + 24 + 15 = 79\)
Step 4
Multiply by 2: \(SA = 2 \times 79 = \mathbf{158 \text{ cm}^2}\)

Example 2 — A cube has edge length 4 cm. Find its surface area.

Step 1
A cube: \(l = w = h = 4\) cm.
Step 2
Each face has area \(4 \times 4 = 16 \text{ cm}^2\).
Step 3
6 identical faces: \(SA = 6 \times 16 = \mathbf{96 \text{ cm}^2}\)
Step 4
Check with formula: \(2(lw + lh + wh) = 2(16+16+16) = 2 \times 48 = 96\) ✓
Formula shortcut for a cube: \(SA = 6s^2\) where \(s\) is the side length. A cube has 6 identical square faces.
✓ Quick Check 1 — Surface Area of Cuboids
Question 1 of 10
Surface Area of a Cylinder

A cylinder has two circular faces and one curved surface. Unrolling the curved surface gives a rectangle, making the formula straightforward.

\[SA_{\text{cylinder}} = 2\pi r^2 + 2\pi r h\]
\(r\) = radius of circular ends, \(h\) = height of cylinder
\(2\pi r^2\) = area of 2 circles  |  \(2\pi r h\) = area of curved surface (rectangle: width \(= 2\pi r\), height \(= h\))
Net of a Cylinder — 2 circles + 1 rectangle (unrolled curved surface) Circle area = πr² r Curved surface unrolled width = 2πr (circumference) area = 2πr × h 2πr h Circle area = πr² SA = 2×πr² (circles) + 2πrh (rectangle) SA of cylinder with r = 3 cm, h = 10 cm 2 circles 2πr² = 2×π×9 = 18π ≈ 56.5 cm² + Curved surface 2πrh = 2×π×3×10 = 60π ≈ 188.5 cm² = Total SA 78π = 18π + 60π ≈ 245.0 cm²

Example 3 — Find the surface area of a cylinder with radius 5 cm and height 12 cm (give answer to 1 d.p.)

Step 1
Area of two circular ends: \(2\pi r^2 = 2 \times \pi \times 5^2 = 2 \times \pi \times 25 = 50\pi\)
Step 2
Area of curved surface: \(2\pi r h = 2 \times \pi \times 5 \times 12 = 120\pi\)
Step 3
Total: \(SA = 50\pi + 120\pi = 170\pi\)
Step 4
\(170\pi \approx 170 \times 3.14159 \approx \mathbf{534.1 \text{ cm}^2}\)

Example 4 — A tin can has radius 4 cm and height 9 cm. How much metal is needed to make it? (1 d.p.)

Step 1
Two circular ends: \(2\pi r^2 = 2 \times \pi \times 16 = 32\pi \approx 100.5 \text{ cm}^2\)
Step 2
Curved surface: \(2\pi r h = 2 \times \pi \times 4 \times 9 = 72\pi \approx 226.2 \text{ cm}^2\)
Step 3
\(SA = 32\pi + 72\pi = 104\pi \approx \mathbf{326.7 \text{ cm}^2}\)
Common error: Forgetting to include both circular ends. The formula \(2\pi r^2\) covers BOTH circles. If you only need the curved surface (e.g. a pipe open at both ends), use \(2\pi r h\) only.
✓ Quick Check 2 — Surface Area of Cylinders
Question 1 of 10
Real-Life Applications and Composite Shapes

Surface area problems in real life often involve painting, wrapping or tiling. Composite shapes require careful thought about which faces are exposed.

Painting a Room — only walls and ceiling, not floor Front wall Side wall Ceiling ✓ Room: 5 m × 4 m × 2.5 m ✓ Front+Back walls: 2×(5×2.5) = 25 m² ✓ Left+Right walls: 2×(4×2.5) = 20 m² ✓ Ceiling: 5×4 = 20 m² ✗ Floor: not painted Total to paint = 25+20+20 = 65 m² At 12 m²/litre → need 65÷12 ≈ 5.5 litres Composite Shape — subtract the joined (hidden) face Big box Small box Joined face (subtract ×2) Composite SA Approach 1. Find SA of each cuboid separately 2. Add them together 3. Subtract 2 × area of joined face (both faces hidden — one on each box) SAₙₒₒₛₖᵢᵗᵉ = SAᾑ + SAᾒ − 2Aₖₒᵢₙᵗ The joined face is hidden on BOTH shapes, so subtract it twice.

Example 5 — A room is 6 m × 4 m × 2.5 m. How much paint is needed if 1 litre covers 10 m² and the floor is not painted?

Step 1
Identify surfaces to paint: 4 walls + ceiling (not floor).
Step 2
Front/back walls: \(2 \times (6 \times 2.5) = 30 \text{ m}^2\)
Step 3
Left/right walls: \(2 \times (4 \times 2.5) = 20 \text{ m}^2\)
Step 4
Ceiling: \(6 \times 4 = 24 \text{ m}^2\). Total = \(30 + 20 + 24 = 74 \text{ m}^2\). Litres needed: \(74 \div 10 = \mathbf{7.4 \text{ litres}}\)

Example 6 — Two cuboids are glued together. The joined face is 3 cm × 2 cm. Cuboid A has SA = 80 cm², Cuboid B has SA = 52 cm². Find the composite SA.

Step 1
Find the area of the joined face: \(3 \times 2 = 6 \text{ cm}^2\).
Step 2
Add individual surface areas: \(80 + 52 = 132 \text{ cm}^2\).
Step 3
Subtract both hidden faces: \(132 - 2 \times 6 = 132 - 12 = \mathbf{120 \text{ cm}^2}\).
Key rule for composites: When two shapes are joined, the touching face is hidden on both shapes. Always subtract it twice from the total of the individual surface areas.

Interactive — Net Unfolder

Use the sliders to set the cuboid dimensions. Toggle between the flat net and the 3D view.

3D isometric view — 3 visible faces shown

Practice: Surface Area

Find the surface area of each shape (in cm²). Type your answer, then click Check.

#QuestionAnswerResult
1Cube with side 3 cm. Find the surface area.
2Cuboid 4 × 3 × 2 cm. Find the surface area.
3Cube with side 5 cm. Find the surface area.
4Cuboid 6 × 2 × 2 cm. Find the surface area.
5Cube with side 2 cm. Find the surface area.
6Cuboid 5 × 4 × 3 cm. Find the surface area.
7Cube with side 10 cm. Find the surface area.
8Cuboid 8 × 5 × 2 cm. Find the surface area.
9Cube with side 4 cm. Find the surface area.
10Cuboid 10 × 6 × 4 cm. Find the surface area.

⚠ Things to Watch Out For

  • Confusing surface area with volume: Surface area is the total area of all the outer faces (measured in square units, e.g. cm²) — volume is the space inside (cubic units, cm³). Don't mix up the formulas.
  • Missing a face: Count every face of the shape carefully — an open-topped container has one fewer face than a fully closed one.
  • Forgetting paired faces: A cuboid has 3 PAIRS of identical faces — calculate one of each pair, then double it, rather than working out all 6 individually.
  • Using diameter instead of radius: The cylinder formula \(SA = 2\pi r^2 + 2\pi rh\) needs the RADIUS. Halve the diameter first if that's what you're given.
  • Mixed units: Convert every measurement to the same unit (e.g. all in cm) before calculating — mixing cm and m gives a wildly wrong answer.
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