Cuboid and cube: surface area and volume
To paint a cardboard box you paint all 6 faces (surface area). Volume tells how much fits inside.
Surface area: total area of all faces (square units). Volume: space inside (cubic units).
In the figure: Cuboid 5 × 4 × 3: volume = 5 × 4 × 3 = 60; surface area = 2 (20 + 12 + 15) = 94.
Formula: Cuboid: surface area = 2 (l × b + b × h + l × h) ; volume = l × b × h
Formula: Cube: surface area = 6 × side × side ; volume = side × side × side
Cuboid and cube: surface area and volume
In the app you can move this figure with your finger.
Open in the app- Easy: l = 5, b = 4, h = 3: volume = 5 × 4 × 3 = 60 cubic cm.
- Medium: l = 10, b = 6, h = 5: surface area = 2 (60 + 30 + 50) = 2 × 140 = 280 sq cm.
- Exam-type: Water tank 2 m × 1.5 m × 1 m. Volume = 2 × 1.5 × 1 = 3 cubic m = 3000 litres (1 cubic m = 1000 litres).
- Exam-type: Cube with side 6 cm: surface area = 6 × 6 × 6 = 216 sq cm ; volume = 6 × 6 × 6 = 216 cubic cm.
Surface area is in square units (cm²), volume in cubic units (cm³).
They are different things, even if the numbers match (cube of side 6), the units differ.
If the side of a cube is known, skip the long formula: surface area = 6 × side × side; volume = side × side × side. Memory clue: Surface area is painting (square units), volume is water (cubic units).
Spot the difference
| Point | Cuboid | Cube |
|---|---|---|
| Length, breadth, height | can all differ | all three equal |
| Surface area | 2 (l × b + b × h + l × h) | 6 × side × side |
| Volume | l × b × h | side × side × side |
| Example | 4, 3, 2: volume 24 | side 3: volume 27 |
Practice questions
- 1Cuboid with l = 9, b = 4, h = 3. Volume?
- 2A cube has side 5 cm. Volume?
- 3A cube has side 3 cm. Surface area?
- 4Cuboid with l = 6, b = 4, h = 3. Surface area?
- 5A tank 3 m × 2 m × 1 m. How many litres of water can it hold?
- 6A box 12 cm × 10 cm × 5 cm. Total area of the four side faces (leaving the top and bottom)?
- 7A cuboid has volume 200 cubic cm, l = 8, b = 5. Its height?
- 8The side of a cube is changed from 2 cm to 4 cm. By how many times does the volume grow?
Show answers (try them yourself first)
- 19 × 4 × 3 = 108 cubic cm.
- 25 × 5 × 5 = 125 cubic cm.
- 36 × 3 × 3 = 54 sq cm.
- 42 (24 + 12 + 18) = 2 × 54 = 108 sq cm.
- 5Volume = 3 × 2 × 1 = 6 cubic m = 6000 litres.
- 62 (12 × 5 + 10 × 5) = 2 (60 + 50) = 220 sq cm.
- 78 × 5 = 40; 200 ÷ 40 = 5 cm.
- 82 × 2 × 2 = 8; 4 × 4 × 4 = 64; 64 ÷ 8 = 8 times.
Cylinder
A tin, a pipe, a glass — all cylinders. Unroll the label on a tin and you get a rectangle: length = circumference of the circle, width = height.
A cylinder has two circular bases and one curved surface. The curved surface is the unrolled rectangle (2πr × h).
In the figure: Cylinder r = 7, h = 10: the curved surface unrolls into a 44 × 10 rectangle (length = circumference 2πr = 44): area = 44 × 10 = 440.
Formula: Curved area = 2 π r h ; total surface area = 2 π r (r + h) ; volume = π r² h
In words: Take π = 22/7. Curved area = circumference × height ; volume = base area × height.
Cylinder
In the app you can move this figure with your finger.
Open in the app- Easy: r = 7, h = 10: curved area = 2 × 22/7 × 7 × 10 = 440 sq cm.
- Medium: r = 7, h = 10: total area = 440 + 2 × (22/7 × 49) = 440 + 308 = 748 sq cm.
- Exam-type: A bucket: r = 14 cm, h = 25 cm. Volume = 22/7 × 14 × 14 × 25 = 15400 cubic cm = 15.4 litres (1000 cubic cm = 1 litre).
Curved area = circumference × height; volume = base area × height. Take π = 22/7 when r is a multiple of 7.
For an open tin (no lid) count the curved area plus just one base.
Both volumes follow one rule: base area × height. Memory clue: Cylinder: "base area times height".
Spot the difference
| Point | Cuboid | Cylinder |
|---|---|---|
| Base | rectangle | circle |
| Volume | base area × height | base area × height |
| Faces | 6 flat | 2 flat + 1 curved |
Practice questions
- 1Cuboid with l = 8, b = 5, h = 2. Volume?
- 2A cube has side 4 cm. Surface area?
- 3Cylinder with r = 7, h = 10. Volume?
- 4Cuboid with l = 10, b = 6, h = 5. Surface area?
- 5A tank 2 m × 1.5 m × 1 m. How many litres of water can it hold?
- 6A tin with r = 7 cm and h = 10 cm. Area of its label (curved surface)?
- 7A cube has volume 125 cubic cm. Find its side and surface area.
- 8A bucket has r = 14 cm, h = 25 cm. How many 2-litre jugs fill it?
Show answers (try them yourself first)
- 18 × 5 × 2 = 80 cubic cm.
- 26 × 4 × 4 = 96 sq cm.
- 31540 cubic cm.
- 42 (60 + 30 + 50) = 280 sq cm.
- 5Volume 3 cubic m = 3000 litres.
- 62 × 22/7 × 7 × 10 = 440 sq cm.
- 7Side 5 (5 × 5 × 5 = 125) ; surface area = 6 × 25 = 150 sq cm.
- 8Volume 15400 cubic cm = 15.4 litres ; 15.4 ÷ 2 = 7.7, so 8 jugs.