Volume (Grade 12 NSC Matric Mathematical Literacy): Revision Notes
Volume
What is volume?
Volume is the amount of space occupied by a 3-dimensional object or container. When you think about volume, imagine filling up a box or bottle with water - the volume tells you how much space is available inside that shape.
Think of volume as the answer to the question: "How much stuff can fit inside this shape?" Whether it's water in a bottle, air in a balloon, or concrete in a foundation, volume measures that capacity.
Volume is always measured using cubic units because we are measuring in three dimensions (length, width, and height). Common cubic units include:
- mm³ (cubic millimetres)
- cm³ (cubic centimetres)
- m³ (cubic metres)
- km³ (cubic kilometres)
Volume formulas for common shapes
Different 3D shapes have specific formulas for calculating their volume. The two most important shapes for your exam are rectangular boxes and cylinders.
Rectangular box (or rectangular prism)
Formula:
Where:
- = length
- = breadth (width)
- = height
To find the volume of a rectangular box, simply multiply all three dimensions together.
Cylinder
Formula:
Where:
- = radius of the circular base
- = height of the cylinder
- ≈ 3.142
Remember that the radius is half the diameter, so if you are given the diameter, divide it by 2 first. This is a common source of errors in cylinder volume calculations.
Important unit conversions
When working with volume problems, you often need to convert between different units:
Key conversion:
This conversion is especially useful when dealing with water capacity, fuel tanks, or swimming pools. Make sure you memorise this relationship!
Problem-solving approach
Follow these steps when solving volume problems:
- Identify the shape - Is it a rectangular box or cylinder?
- Write down the given measurements - Check all dimensions are in the same units
- Choose the correct formula - Use the appropriate volume formula
- Substitute the values - Replace letters with numbers in the formula
- Calculate carefully - Use a calculator for π calculations
- Check your units - Make sure your answer has cubic units
- Convert if necessary - Change units if the question requires it
Following this systematic approach will help you avoid common mistakes and ensure you get full marks for showing your method clearly.
Worked examples
Worked Example 1: Rectangular foundation volume
A house foundation has dimensions 8 m × 0.5 m × 0.5 m. Calculate the volume.
Solution: Using
Worked Example 2: Cylindrical container volume
A cylindrical biscuit has diameter 80 mm and height 7 mm. Calculate its volume.
Solution: First convert diameter to radius:
Using
Worked Example 3: Swimming pool volume with conversions
A swimming pool measures 15 m × 5 m × 1.3 m. Calculate the volume and convert to litres.
Solution: Using
Converting to litres:
Worked Example 4: Time calculation
If a pump fills 2 litres per second, how long to fill the pool from Example 3?
Solution: Pool volume = 97500 litres Rate = 2 litres per second
Time =
Converting to hours and minutes:
Worked Example 5: Cost calculation
If water costs R8.64 per kilolitre, what is the cost to fill the pool?
Solution: Pool volume = 97.5 m³ = 97.5 kilolitres
Cost =
Exam tips
Essential exam strategies:
- Always check your units - Make sure all measurements are in the same unit before calculating
- Remember conversions - Know that 1 m³ = 1000 litres
- Use brackets carefully - When calculating , square the radius before multiplying by π
- Don't round too early - Keep full accuracy until your final answer
- Show your working - Write down the formula first, then substitute values
Key Points to Remember:
- Volume measures the space inside a 3D object and is always expressed in cubic units
- Rectangular box volume = length × breadth × height ()
- Cylinder volume = π × radius² × height ()
- Key conversion: 1 m³ = 1000 litres = 1 kilolitre
- Problem-solving strategy: Identify shape → Check units → Apply formula → Calculate → Convert if needed