Perimeter, area and volume

Definition of Perimeter

Perimeter is the total length of the outside of a shape or the continuous line forming the boundary of a closed geometric figure.
It is calculated by adding together the lengths of each side.
Units of measurement: millimetres (mm), centimetres (cm), metres (m), or kilometres (km).

Perimeter Formulae for Common Shapes

Shape	Formula
Rectangle	$P = 2 \times \text{length} + 2 \times \text{width}$
Square	$P = 4 \times \text{side}$
Triangle	$P = \text{side}_1 + \text{side}_2 + \text{side}_3$
Circle (circumference)	$P=2π×radiusP = 2\pi \times \text{radius}$

Measuring Perimeter

For rectangles, squares, and triangles, use a ruler to measure each side and sum the values.
For circles, use string to trace the outline, then measure the string length using a ruler.
The perimeter of a circle is called its circumference.

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Worked Example 1: Calculating Perimeter

Question:

Mr. and Mrs. Dlamini have a rectangular backyard with a length of 5 cm and width of 4.2 cm in a diagram. What is the actual perimeter in metres if the diagram scale is 1:100?

Solution:

Calculate the perimeter using the rectangle formula:

P = 2(5) + 2(4.2) = 18.4 \text{ cm}

Convert using the scale:

18.4 \text{ cm} \times 100 = 18.4 \text{ m}

Final Answer: 18.4 m

Worked Example 2: Applying Scale to Find Actual Size

Mrs Dlamini wants to dig up some of the lawn and plant a triangular vegetable garden as shown in the diagram alongside

The scale given is 1:100, meaning 1 cm on the diagram represents 100 cm in reality.

b) Converting to Actual Perimeter

12 \text{ cm} \times 100 = 1 200 \text{ cm}

Since 1 200 cm = 12 m, the actual perimeter of the garden is 12 metres.

Area

Definition of Area

Area is the measure of the surface enclosed by a shape.
Units of measurement: square millimetres (mm²), square centimetres (cm²), square metres (m²), square kilometres (km²).

Area Formulae for Common Shapes

Shape	Formula
Rectangle	$A = \text{length} \times \text{width}$
Square	$A = \text{side}^2$
Triangle	$A = \frac{1}{2} \times \text{base} \times \text{height}$
Circle	$A = \pi \times \text{radius}^2$

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Worked Example 2: Calculating Area

Question:

Calculate the area of a triangular garden with a base of 4 cm and a height of 12 cm.

Solution:

$A = \frac{1}{2} \times 4 \times 12$

$= 24 \text{ cm}^2$

Final Answer: 24 cm²

Combining Areas of Different Shapes

Sometimes, complex shapes are formed by combining multiple basic shapes.
Steps: 3. Break the shape into simpler parts (e.g., rectangle + triangle). 4. Find the area of each shape separately. 5. Add or subtract areas accordingly.

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Worked Example 3: Combining Shapes

Question:

A rare coin has a circular shape with a square cut-out.

The circle has a diameter of 3 cm.
The square has a side length of 0.9 cm.
Find the remaining area of the coin.

Solution:

Calculate the area of the circle:

A \ circle= \pi \times \left(\frac{3}{2}\right)^2

= 3.142 \times 2.25 = 7.07 \text{ cm}^2

Calculate the area of the square:

A\ square = 0.9 \times 0.9 = 0.81 \text{ cm}^2

Subtract the square from the circle:

A\ coin = 7.07 - 0.81 = 6.26 \text{ cm}^2

Final Answer: 6.26 cm²

Volume

Definition of Volume

Volume is the amount of space occupied by a 3D object.
Units of measurement: cubic millimetres (mm³), cubic centimetres (cm³), cubic metres (m³).
Key conversion: $:highlight[1 m^3 = 1000 \text{ L}]$

Volume Formulae for Common Shapes

Shape	Formula
Rectangular box	$V = \text{length} \times \text{width} \times \text{height}$
Cylinder	$V = \pi \times \text{radius}^2 \times \text{height}$

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Worked Example 4: Calculating Volume

Question:

Cedric is building a rectangular foundation with dimensions 8 m × 0.5 m × 0.5 m.

Calculate the volume and cost of concrete if it costs R180/m³.

Solution:

Calculate volume: $V = 8 \times 0.5 \times 0.5 = 2 \text{ m}^3$
Find total cost: $2 \times R180 = R360$

Final Answer: R360

4. Multi-Step Problems

Complex problems require multiple steps, including unit conversions, cost calculations, or combining shapes.
Example: Swimming Pool Calculation
- Length = 15 m, Depth = 1.3 m, Width = 5 m.
- Convert to litres: $:highlight[1 m^3 = 1000 \text{ L}]$
- Find time to fill the pool (using a pump rate of 2 L/sec). Solution Summary:

Calculate volume: $15 \times 1.3 \times 5 = 97.5 \text{ m}^3$
Convert to litres: $97.5 \times 1000 = 97 500 \text{ L}$
Calculate time: $97 500 \div 2 = 48 750 \text{ seconds} = :success[13 \text{ hours } 32 \text{ min}]$

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Final Summary

Perimeter: Sum of all sides.
Area: Space inside a shape.
Volume: Space inside a 3D object.
Use correct formulae for each shape.
Apply real-world problem-solving using conversion factors, cost calculations, and combined shapes.

Perimeter, area and volume Simplified Revision Notes for NSC Mathematical Literacy

Perimeter, area and volume

Definition of Perimeter

Perimeter Formulae for Common Shapes

Measuring Perimeter

Worked Example 1: Calculating Perimeter

Worked Example 2: Applying Scale to Find Actual Size

b) Converting to Actual Perimeter

Area

Definition of Area

Area Formulae for Common Shapes

Worked Example 2: Calculating Area

Combining Areas of Different Shapes

Worked Example 3: Combining Shapes

Volume

Definition of Volume

Volume Formulae for Common Shapes

Worked Example 4: Calculating Volume

4. Multi-Step Problems

Final Summary

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