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Perimeter and Area of Shapes
Key Formulas for Perimeter and Area
Rectangle
Perimeter: P = 2(l + w)
Area: A = l × w
Where l is length, w is width
Triangle
Perimeter: P = a + b + c
Area: A = ½ × b × h
Where b is base, h is height
Circle
Circumference: C = 2πr = πd
Area: A = πr²
Where r is radius, d is diameter
Trapezium
Perimeter: Sum of all sides
Area: A = ½(a + b)h
Where a and b are parallel sides, h is height
Worked Examples
Example 1: Rectangular Field
Problem: A rectangular field measures 150m by 90m. Calculate its perimeter and area.
Solution:
Perimeter = 2(l + w) = 2(150 + 90) = 2 × 240 = 480m
Area = l × w = 150 × 90 = 13,500m²
Example 2: Triangular Plot
Problem: A triangular plot has sides of 120m, 90m, and 150m. The height corresponding to the 150m base is 72m. Find its area.
Solution:
Area = ½ × base × height = ½ × 150 × 72 = 5,400m²
Example 3: Circular Garden
Problem: A circular garden has a radius of 14m. Calculate its circumference and area.
Solution:
Circumference = 2πr = 2 × 3.142 × 14 ≈ 87.98m
Area = πr² = 3.142 × 14² ≈ 3.142 × 196 ≈ 615.83m²
[Diagram: Composite shape made of rectangle and semicircle]
Problem-Solving Strategies
1. Identify Components
Break down complex shapes into simpler ones (rectangles, triangles, circles).
2. Find Missing Dimensions
Use properties of shapes and algebra to find unknown lengths.
3. Unit Consistency
Ensure all measurements are in the same units before calculating.
4. Check Reasonableness
Estimate whether your answer makes sense in context.
Real-World Application: Fencing a Field
A field has the shape of a trapezium with dimensions: AB = 150m, BC = 90m, CD = 120m, with right angles at B and C. Calculate the length of fencing needed to enclose the field and the cost if fencing costs $48 per 5-meter section.
Solution approach:
1. Find the length of AD using the Pythagorean theorem
2. Calculate the total perimeter
3. Determine how many 5m sections are needed
4. Multiply by cost per section
Practice Quiz
Question 1
Calculate the area of a trapezium with parallel sides of 8cm and 12cm, and height 5cm.
Question 2
A rectangular garden measures 15m by 10m. What is the perimeter?
Question 3
A circular pond has a diameter of 7m. Calculate its circumference and area.
Question 4
A triangular plot has sides 90m, 120m, and 150m. Show that it is a right-angled triangle and find its area.
Question 5
A composite shape consists of a rectangle (20m × 15m) with a semicircle (diameter 15m) on one end. Calculate the total area.
Quiz Answers
- Area = ½(8 + 12) × 5 = ½ × 20 × 5 = 50cm²
- Perimeter = 2(15 + 10) = 2 × 25 = 50m
- Circumference = π × 7 ≈ 21.99m, Area = π × (3.5)² ≈ 38.48m²
- Since 90² + 120² = 8100 + 14400 = 22500 = 150², it is right-angled. Area = ½ × 90 × 120 = 5400m²
- Rectangle area = 20 × 15 = 300m², Semicircle area = ½ × π × (7.5)² ≈ 88.36m², Total ≈ 388.36m²
Conversion Reference
| Measurement | Conversion | Example |
|---|---|---|
| Length | 1km = 1000m, 1m = 100cm | 2.5km = 2500m |
| Area | 1m² = 10,000cm² | 5m² = 50,000cm² |
| Hectares | 1 hectare = 10,000m² | 85,000m² = 8.5 hectares |
Exam Tips
Show Your Working
Even if your final answer is wrong, you may get marks for correct method.
Include Units
Always include units in your final answer (cm, m, cm², m², etc.).
Use Formulas Correctly
Write down the formula first, then substitute values.
Check Calculations
If time allows, recheck your calculations for errors.
Common Mistake to Avoid
When calculating the perimeter of a shape, make sure you include all the sides. For composite shapes, be careful not to double-count or miss any sides.