Volume & Surface Area
Calculate the volume and surface area of rectangular prisms and understand the difference between 2D and 3D measurement.
Volume of a Rectangular Prism
Volume is the amount of 3D space inside a solid shape. For a rectangular prism (box), it is measured in cubic units (cm³, m³, mm³).
Formula
V = length × width × height
Also written as V = l × w × h
Visual: Counting Unit Cubes
Imagine filling a box with 1 cm³ cubes. A box 4 cm × 3 cm × 2 cm holds:
12 cubes per layer × 2 layers = 24 cm³
Formula check: 4 × 3 × 2 = 24 cm³ ✓
Surface Area of a Rectangular Prism
Surface area is the total area of all the faces (flat surfaces) of a 3D shape. It is measured in square units (cm², m²). Think of unfolding the box flat — that flat shape is called a net.
A rectangular prism has 6 faces in 3 pairs of opposite, identical rectangles:
Top & Bottom
2 × (l × w)
Front & Back
2 × (l × h)
Left & Right
2 × (w × h)
Total Surface Area Formula:
SA = 2(lw + lh + wh)
Units of Measurement
Choosing the right unit is important. Here are the common volume units and their conversions:
| Unit | Symbol | Conversion |
|---|---|---|
| Cubic millimetre | mm³ | 1 cm³ = 1000 mm³ |
| Cubic centimetre | cm³ | 1 L = 1000 cm³ |
| Cubic metre | m³ | 1 m³ = 1 000 000 cm³ |
Key Vocabulary
Volume
The amount of 3D space inside a solid shape. Measured in cubic units (cm³, m³).
Surface Area
The total area of all faces of a 3D shape. Measured in square units (cm², m²).
Rectangular Prism
A 3D solid with 6 rectangular faces. Also called a cuboid. A cube is a special prism where all sides are equal.
Net
A 2D shape that can be folded to make a 3D solid. The net of a rectangular prism shows all 6 faces laid flat.
Worked Examples
Find the volume of a box 5 cm × 3 cm × 4 cm.
Formula: V = l × w × h
Substituting: V = 5 × 3 × 4 = 60 cm³
Find the surface area of a rectangular prism 6 m × 4 m × 3 m.
Formula: SA = 2(lw + lh + wh)
Step 1: lw = 6 × 4 = 24 m²
Step 2: lh = 6 × 3 = 18 m²
Step 3: wh = 4 × 3 = 12 m²
Step 4: SA = 2(24 + 18 + 12) = 2 × 54 = 108 m²
A cube has volume 125 cm³. What is its side length?
For a cube: V = s³ (side cubed)
So: s³ = 125
Therefore: s = ³√125 = 5 cm
Check: 5 × 5 × 5 = 125 ✓
Knowledge Check
Loading questions…
Key Concepts Summary
- ●Volume of a rectangular prism: V = l × w × h. Units are cubic (cm³, m³).
- ●Surface area = total area of all 6 faces: SA = 2(lw + lh + wh). Units are square (cm², m²).
- ●A rectangular prism has 6 rectangular faces in 3 matching pairs.
- ●A cube is a special prism where all sides are equal: V = s³, SA = 6s².
- ●1 cm³ = 1 mL, so 1000 cm³ = 1 litre. This connects volume to capacity.