Area and Perimeter
From painting a room to fencing a garden, area and perimeter are essential for countless real-world tasks. This guide will walk you through the key formulas for calculating the distance around a shape (perimeter) and the space it covers (area).
Perimeter vs. Area
Perimeter: The Distance Around
Perimeter is the total distance around the outside edge of a 2D shape. To find it, add up the lengths of all the sides.
Measured in units of length: cm, m, km.
Area: The Space Inside
Area is the amount of surface a 2D shape covers. Each type of shape has its own formula.
Measured in square units: cm², m², km².
Key Area Formulas
Area of a Triangle
Area of a Parallelogram
Area of a Trapezium
Here $a$ and $b$ are the lengths of the two parallel sides and $h$ is the perpendicular height between them.
Worked Example: Composite Shape
Find the total area of the shape below.
Split the shape into a rectangle and a triangle.
- Area of Rectangle:
$10 \times 6 = 60 \text{ cm}^2$. - Area of Triangle:
Base = 10 cm, height = 4 cm.
$\frac{1}{2} \times 10 \times 4 = 20 \text{ cm}^2$. - Total Area:
$60 + 20 = 80 \text{ cm}^2$.
Tutor Insights
🤔 Common Misunderstandings
- Perpendicular vs. slant height: Using a slanted side instead of the perpendicular height in the area formula for a triangle or parallelogram.
- Confusing Area and Perimeter: Calculating one when the question asks for the other.
- Forgetting to halve the result for the area of a triangle or trapezium.
📝 Common Exam Mistakes
- Using the wrong formula for the shape.
- Including internal lines when calculating the perimeter of a composite shape.
- Forgetting units (cm or m for perimeter; cm² or m² for area) in the final answer.
Practice Questions
- A rectangle has a length of 12 cm and a width of 5 cm. Calculate its perimeter and area.
- Find the area of a triangle with a base of 10 m and a perpendicular height of 6 m.
- Calculate the area of a trapezium with parallel sides of 8 cm and 12 cm, and a perpendicular height of 5 cm.
- A triangle has an area of 48 cm² and a base of 16 cm. What is its perpendicular height?
Show Answers
- Perimeter: $(12+5) \times 2 = 34 \text{ cm}$. Area: $12 \times 5 = 60 \text{ cm}^2$.
- Working: $\frac{1}{2} \times 10 \times 6 = 30$.
Answer: 30 m². - Working: $\frac{1}{2} \times (8+12) \times 5 = \frac{1}{2} \times 20 \times 5 = 50$.
Answer: 50 cm². - Working: $48 = \frac{1}{2} \times 16 \times h \implies h = 48 \div 8 = 6$.
Answer: 6 cm.
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