Complete Notes on Area - Grade 5
1. Area of Squares and Rectangles
Definition
The area is the amount of space inside a shape. It is measured in square units (like cm², m², in²).
Square
\[ \text{Area} = \text{side} \times \text{side} = s^2 \]
Rectangle
\[ \text{Area} = \text{length} \times \text{width} = l \times w \]
Examples
- A square with side = 7 cm: \( 7 \times 7 = 49 \) cm²
- A rectangle 8 m × 5 m: \( 8 \times 5 = 40 \) m²
✓ Label your answer with square units (cm², m², etc.)
2. Area of Rectangles with Fractions
Fractions as Side Lengths
Use the same formula: \[ A = l \times w \] even if lengths are fractions!
Examples
- \( \frac{3}{4} \) m × \( \frac{2}{5} \) m
Area = \( \frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10} \) m² - \( 1\frac{1}{2} \) in × \( 2\frac{2}{3} \) in
First convert to improper fractions:
\( 1\frac{1}{2} = \frac{3}{2}, 2\frac{2}{3} = \frac{8}{3} \)
\( \frac{3}{2} \times \frac{8}{3} = \frac{24}{6} = 4 \) in²
✓ Always simplify your answer.
3. Area of Rectangles with Fractions and Mixed Numbers
For mixed numbers, break into whole + fraction or convert to improper fractions.
Multiply as usual.
Examples
- If a rectangle has sides \( 2\frac{1}{4} \) cm × \( 1\frac{2}{3} \) cm:
Convert: \( 2\frac{1}{4} = \frac{9}{4} \), \( 1\frac{2}{3} = \frac{5}{3} \)
Area = \( \frac{9}{4} \times \frac{5}{3} = \frac{45}{12} = 3\frac{9}{12} = 3\frac{3}{4} \) cm² - Break into parts:
\( 1\frac{1}{2} = 1 + \frac{1}{2} \), \( 2\frac{1}{3} = 2 + \frac{1}{3} \)
Multiply parts and add: \( 1 \times 2 = 2, 1 \times \frac{1}{3} = \frac{1}{3}, \frac{1}{2} \times 2 = 1, \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \)
Add: \( 2 + \frac{1}{3} + 1 + \frac{1}{6} = 3 + \frac{1}{3} + \frac{1}{6} = 3 + \frac{1}{2} = 3\frac{1}{2} \) units²
4. Area of Compound Figures
Definition
Compound (or composite) figures are made of two or more simple shapes. Find the area of each shape and add (or subtract) their areas.
Examples:
- A figure made of 2 rectangles:
Rectangle 1: \( l_1 \times w_1 \)
Rectangle 2: \( l_2 \times w_2 \)
Total area = \( (l_1 \times w_1) + (l_2 \times w_2) \) - Subtract if necessary (example: a hole):
Area = Area of big shape − Area of hole
✓ Add or subtract each area as needed.
5. Area Between Two Rectangles
Find area of big rectangle, then smaller rectangle.
Area between = big rectangle − small rectangle.
\[ \text{Area}_{\text{big}} = l_B \times w_B \]
\[ \text{Area}_{\text{small}} = l_S \times w_S \]
\[ \text{Area between} = \text{Area}_{\text{big}} - \text{Area}_{\text{small}} \]
Example:
- Big rectangle: 12 × 9 = 108 units²; Small rectangle: 5 × 6 = 30 units²; Area between = 108 − 30 = 78 units²
6. Area of Figures on Grids
Definition
Count Unit Squares within the shape.
If between grid lines, use fractions.
Example:
- 5 squares inside figure = 5 units²
- If shape covers half a square: area = \( \frac{1}{2} \) units²
✓ For partial squares, count as fractions.
7. Area and Perimeter: Word Problems
How to Solve
- Read problem carefully
- Identify what shape or shapes are involved
- Write down known values
- Select appropriate formulas (area or perimeter)
- Calculate as required; answer in correct units
Example:
- Find the area and perimeter of a 6 m × 4 m rectangle:
Area: \( 6 \times 4 = 24 \) m²
Perimeter: \( 2 \times (6+4) = 20 \) m
✓ Use correct formula for each.
8. Area and Perimeter with Fractions & Decimals
Use the same process: Multiply for area, add for perimeter. Fractions and decimals just require careful calculation and simplification.
Example:
- Area: \( 2.5 \) m × \( 1.2 \) m = \( 3.0 \) m²
- Area: \( \frac{2}{3} \) ft × \( \frac{3}{4} \) ft = \( \frac{6}{12} = \frac{1}{2} \) ft²
- Perimeter: \( 1.2 + 2.5 + 1.2 + 2.5 = 7.4 \) m
📚 Quick Reference Formula Sheet
Square
\[ A = s^2 \]
where \(s\) = side
Rectangle
\[ A = l \times w \]
where \(l\) = length, \(w\) = width
Compound Shape
\[ A = \text{Area}_1 + \text{Area}_2 + ... \] or \( \text{Area}_1 - \text{Area}_2 \) if subtracting
Grid (Count Squares)
\[ A = \text{Number of unit squares} \]