📦 Surface Area

Cambridge Lower Secondary · Grade 7 · Geometry & Measure

Cuboid

SA = 2(lb + bh + lh)
Add all six rectangular faces

Cylinder

SA = 2πr² + 2πrh
Two circles + curved rectangle

Key Idea

Unfold the net → find every face area → add them all up! 🎁

Watch a cuboid net fold! 📦

What you'll learn:

What surface area means (total area of all faces)
Nets — unfolding 3D shapes to see every face
Surface area of a cuboid: 2(lb + bh + lh)
Surface area of a triangular prism
Surface area of a cylinder: 2πr² + 2πrh
Surface area of a cube: 6s²
Converting area units: cm² ↔ m²

Created by Mr Josh

📖 Learn: Surface Area

Part 1: What is Surface Area?

The surface area of a 3D shape is the total area of all its outer faces added together.

Imagine wrapping a present in wrapping paper — the amount of paper you need is the surface area of the box! 🎁

Surface area is measured in square units: mm², cm², m²
Every flat face contributes its own area to the total
For curved shapes (like cylinders), we include the curved surface too

💡 Tip: Always write the unit squared (e.g. cm²) in your answer!

Part 2: Nets — Unfolding 3D Shapes

A net is what you get when you unfold a 3D shape and lay it completely flat. Every face of the 3D shape becomes a flat 2D shape in the net.

Cuboid net: 6 rectangles (3 pairs of opposite faces)

Cube net: 6 identical squares

Triangular prism net: 2 triangles + 3 rectangles

Cylinder net: 2 circles + 1 rectangle (the curved surface unrolls!)

💡 Remember: Not every cross arrangement of squares is a valid net. Valid nets must fold into a closed shape with no gaps or overlaps.

Part 3: Surface Area of a Cuboid

A cuboid has 6 rectangular faces in 3 pairs:

Top & Bottom: each has area = l × b
Front & Back: each has area = l × h
Left & Right: each has area = b × h

SA = 2(lb + lh + bh)

Where l = length, b = breadth (width), h = height

Cube: All sides equal (s), so SA = 6s²

💡 A cube is a special cuboid where l = b = h = s, giving SA = 2(s·s + s·s + s·s) = 6s²

Part 4: Surface Area of a Triangular Prism

A triangular prism has 5 faces:

2 triangular ends (the "bases")
3 rectangular faces (one for each edge of the triangle)

If the triangle has base b, height h, and the three sides are a, b, c, and the prism length is L:

SA = 2 × (½ × b × h) + (a + b + c) × L

💡 The three rectangles each have width = length of that triangle side, and height = prism length L.

Part 5: Surface Area of a Cylinder & Unit Conversion

A cylinder has 3 surface parts:

2 circular ends: each has area πr²
1 curved surface: unrolls into a rectangle with width = 2πr and height = h

SA = 2πr² + 2πrh

Where r = radius, h = height of cylinder

Converting area units:

1 m = 100 cm, so 1 m² = 100 × 100 = 10 000 cm²
To convert cm² → m²: divide by 10 000
To convert m² → cm²: multiply by 10 000
1 cm = 10 mm, so 1 cm² = 100 mm²

💡 Key: When converting lengths you multiply/divide once; for areas you multiply/divide by the square of the conversion factor!

Created by Mr Josh

💡 Worked Examples

Example 1: Surface Area of a Cuboid

Find the surface area of a cuboid with l = 8 cm, b = 5 cm, h = 3 cm.

Step 1: Identify the three pairs of faces.
Top/Bottom: 8 × 5 = 40 cm²  (×2 = 80)
Front/Back: 8 × 3 = 24 cm²  (×2 = 48)
Left/Right: 5 × 3 = 15 cm²  (×2 = 30)

Step 2: Use the formula.
SA = 2(lb + lh + bh) = 2(40 + 24 + 15) = 2 × 79

Step 3: SA = 158 cm²

✅ Check: 6 faces, 3 pairs — always write units squared!

Example 2: Surface Area of a Cube

A cube has side length s = 4 cm. Find its surface area.

SA = 6s² = 6 × 4² = 6 × 16 = 96 cm²

✅ All 6 faces are identical squares of area s².

Example 3: Surface Area of a Triangular Prism

A triangular prism has a right-angled triangle cross-section with legs 3 cm and 4 cm, hypotenuse 5 cm. The prism length is 10 cm.

Step 1: Area of each triangular face = ½ × 3 × 4 = 6 cm²
Two triangles: 2 × 6 = 12 cm²

Step 2: Three rectangular faces:
3 × 10 = 30 cm²
4 × 10 = 40 cm²
5 × 10 = 50 cm²

Step 3: SA = 12 + 30 + 40 + 50 = 132 cm²

Example 4: Surface Area of a Cylinder

A cylinder has radius r = 5 cm and height h = 12 cm. Find the surface area (use π ≈ 3.142).

Step 1: Two circular ends: 2πr² = 2 × π × 5² = 2 × π × 25 = 50π ≈ 157.1 cm²

Step 2: Curved surface: 2πrh = 2 × π × 5 × 12 = 120π ≈ 376.99 cm²

Step 3: SA = 50π + 120π = 170π ≈ 534.1 cm² (3 s.f.)

✅ Tip: Keep answer in terms of π until the final step for accuracy.

Created by Mr Josh

🎨 Visualizer

📦 Net Folder — Watch it fold!

Press Play to watch a cuboid net fold into a 3D box and back again.

Size: 40

Net is unfolded — press Play to fold it up!

🖌️ Face Painter — Click faces to paint!

Click each face of the cuboid to "paint" it. The total painted area updates as you go.

Face	Area	Painted?

Total painted:
0 cm²

Full SA:
0 cm²

l: 6 cm b: 4 cm h: 3 cm

📐 Prism Explorer — Live net & formula!

Adjust the sliders — the net redraws and the surface area updates in real time.

Length (l):6 cm

Width (b):4 cm

Height (h):3 cm

SA = 2(lb + lh + bh) = 2(24 + 18 + 12) = 108 cm²

🥫 Cylinder Unwrapper — Watch it unroll!

See how the curved surface of a cylinder unrolls into a rectangle labelled 2πr × h.

Radius (r):3 cm

Height (h):7 cm

SA = 2πr² + 2πrh = 56.5 + 131.9 = 188.5 cm²

Created by Mr Josh

✏️ Exercise 1: Identifying Nets

Decide whether each description is a valid net for the given shape.

1. A cross-shaped arrangement of 6 equal squares. Is this a valid net of a cube?

2. A row of 6 squares in a straight line. Is this a valid net of a cube?

3. A net has 2 circles and 1 rectangle. What shape does it fold into?

4. A net has 2 triangles and 3 rectangles. What 3D shape does it make?

5. A net has 6 rectangles, not all the same size. What shape could it fold into?

6. A cube of side 5 cm. How many faces does its net have?

7. True or False: A valid net of a cuboid has 3 pairs of identical rectangles.

8. A triangular prism has how many faces in total?

Created by Mr Josh

✏️ Exercise 2: Surface Area of Cuboids

Use SA = 2(lb + lh + bh). Give answers in cm².

1. Cuboid: l = 4, b = 3, h = 2 cm. SA = cm²

2. Cuboid: l = 7, b = 5, h = 4 cm. SA = cm²

3. Cuboid: l = 10, b = 6, h = 3 cm. SA = cm²

4. Cuboid: l = 9, b = 9, h = 2 cm. SA = cm²

5. Cube: s = 5 cm. SA = cm²

6. Cube: s = 8 cm. SA = cm²

7. Cuboid: l = 12, b = 4, h = 5 cm. SA = cm²

8. Cuboid: l = 6, b = 6, h = 6 cm (it's a cube!). SA = cm²

9. A box has l = 15, b = 8, h = 2 cm. SA = cm²

10. A brick: l = 20, b = 10, h = 6 cm. SA = cm²

Created by Mr Josh

✏️ Exercise 3: Surface Area of Triangular Prisms

Each prism has a right-angled triangle cross-section unless stated. Give answers in cm².

1. Triangle: base 3, height 4, hypotenuse 5 cm. Prism length 8 cm. SA = cm²

2. Triangle: base 6, height 8, hypotenuse 10 cm. Prism length 12 cm. SA = cm²

3. Triangle: base 5, height 12, hypotenuse 13 cm. Prism length 9 cm. SA = cm²

4. Triangle: base 8, height 6, hypotenuse 10 cm. Prism length 15 cm. SA = cm²

5. Equilateral triangle sides all 6 cm, height ≈ 5.196 cm. Prism length 10 cm. SA = cm² (round to 1 d.p.)

Created by Mr Josh

✏️ Exercise 4: Surface Area of Cylinders

Use SA = 2πr² + 2πrh. Use π = 3.142. Round to 1 d.p. unless stated.

1. r = 3 cm, h = 7 cm. SA = cm²

2. r = 5 cm, h = 10 cm. SA = cm²

3. r = 4 cm, h = 4 cm. SA = cm²

4. r = 7 cm, h = 12 cm. SA = cm²

5. Diameter = 8 cm, h = 5 cm. SA = cm²

6. r = 2 cm, h = 9 cm. SA = cm²

Created by Mr Josh

🔀 Exercise 5: Mixed & Unit Conversion

Mixed shapes and unit conversions. Show your units clearly!

1. Convert 25 000 cm² to m². Answer = m²

2. Convert 3.5 m² to cm². Answer = cm²

3. Convert 450 cm² to mm². Answer = mm²

4. Convert 72 000 mm² to cm². Answer = cm²

5. Cuboid: l = 1.2 m, b = 0.8 m, h = 0.5 m. SA = m²

6. Cylinder: r = 0.5 m, h = 2 m. SA = m² (3 s.f.)

7. A cube's SA = 294 cm². What is its side length? s = cm

8. A room (cuboid) is 5 m × 4 m × 3 m. What is the total SA of the 4 walls and ceiling (not floor)? SA = m²

Created by Mr Josh

📝 Practice: 20 Questions

Full mix of all shapes. Give numerical answers; use π = 3.142 for cylinders, round to 1 d.p.

1. Cube s = 3 cm. SA = cm²

2. Cuboid 5 × 3 × 2. SA = cm²

3. Cylinder r = 2, h = 6. SA = cm²

4. Triangular prism: right triangle 3–4–5, length 10. SA = cm²

5. Convert 80 000 cm² → m². = m²

6. Cube s = 7 cm. SA = cm²

7. Cuboid 9 × 4 × 3. SA = cm²

8. Cylinder r = 6, h = 8. SA = cm²

9. Prism: triangle 5–12–13, length 7. SA = cm²

10. Convert 2.4 m² → cm². = cm²

11. Cuboid 11 × 2 × 2. SA = cm²

12. Cylinder diameter = 10, h = 4. SA = cm²

13. Cube: each face area = 49 cm². SA = cm²

14. Prism: triangle 6–8–10, length 5. SA = cm²

15. Convert 350 mm² → cm². = cm²

16. Cuboid 3 × 3 × 8. SA = cm²

17. Cylinder r = 1.5, h = 10. SA = cm²

18. Prism: triangle 8–15–17, length 6. SA = cm²

19. A cuboid SA = 148 cm². If l = 5, b = 4, find h. h = cm

20. Cylinder: SA = 2πr(r + h). If r = 3, SA = 150.8 cm², find h (1 d.p.). h = cm

Created by Mr Josh

🏆 Challenge: 8 Hard Questions

Multi-step and problem-solving. Show all working! Use π = 3.142.

1. A cube's SA is doubled. What happens to the side length? New length = × original

2. Cuboid 8×5×3. Each face is painted at 0.05 cm thickness. Volume of paint = SA × 0.05. Volume = cm³

3. A cylinder has r = h. SA = 2πr² + 2πr² = 4πr². If SA = 314.2 cm², find r (1 d.p.). r = cm

4. A cuboid has l:b:h = 3:2:1 and SA = 352 cm². Find each dimension. l = b = h = cm

5. Two cubes of side 4 cm are joined face-to-face. SA of combined shape = cm²

6. Paint costs £2 per 100 cm². A cylinder (r=5, h=8) needs 2 coats. Total cost = £

7. A triangular prism has SA = 312 cm². The triangle is right-angled with legs 6 cm and 8 cm (hypotenuse 10 cm). Find the prism length. L = cm

8. A hollow cylinder (open at both ends): SA = just the curved surface = 2πrh. If r = 4 cm and SA = 301.6 cm², find h (1 d.p.). h = cm

Created by Mr Josh