Chapter focus
Area measures the surface covered by a two-dimensional shape. Choose the correct formula, use perpendicular heights and keep square units consistent.
These original examples support revision. Complete the full visual investigations and exercises in your assigned Ganita Prakash Part II.
1. Area and units
Area counts how many square units cover a region without gaps or overlaps. If a rectangle is 5 cm long and 3 cm wide, it covers 5 × 3 = 15 square centimetres, written 15 cm².
Area is different from perimeter, which measures the boundary length. A length is measured in units such as cm; area is measured in square units such as cm². Convert units before adding or comparing areas: 1 m² = 10,000 cm² because 1 m = 100 cm and 100 × 100 = 10,000.
2. Common area formulas
- Rectangle: A = length × breadth.
- Square: A = side × side = side².
- Triangle: A = ½ × base × perpendicular height.
- Parallelogram: A = base × perpendicular height.
- Trapezium: A = ½ × (sum of parallel sides) × perpendicular distance between them.
The height used with a base is the perpendicular distance to that base. It is not necessarily a sloping side.
3. Derive and use the triangle formula
Two congruent copies of a triangle can form a parallelogram with the same base and height. The parallelogram’s area is base × height; one triangle is half of it. Therefore A = ½bh.
Worked example
A triangle has base 10 cm and perpendicular height 7 cm. Its area is ½ × 10 × 7 = 35 cm². If 7 cm were a sloping side and not the perpendicular height, this calculation would not be justified.
4. Find area of a composite figure
Split a composite shape into rectangles, triangles or other familiar shapes. Find each area, then add non-overlapping parts or subtract a cut-out. Mark lengths clearly and convert them to one unit first.
Worked example: rectangle with a triangular roof
A 12 m by 8 m rectangle has a triangle on top with base 12 m and perpendicular height 3 m. Total area = rectangle area + triangle area = (12 × 8) + (½ × 12 × 3) = 96 + 18 = 114 m².
If a triangular corner is cut from a rectangle, subtract that triangle’s area instead of adding it.
5. Check whether an answer is reasonable
Estimate by rounding dimensions, check that the answer has square units, and compare it with a surrounding rectangle. A triangle inside a rectangle with the same base and height should have half that rectangle’s area.
For example, a 9 m by 4 m rectangle has area 36 m², so a triangle with the same base and perpendicular height has area 18 m²—not 36 m².
Common errors to avoid
- Do not report perimeter in square units or area in linear units.
- For a triangle, remember the factor 1/2.
- Height must meet the chosen base at 90°.
- Convert all dimensions to a common unit before multiplying or combining areas.
Practice: include units
1. Find the area of a 12 cm by 8 cm rectangle.
Answer: 12 × 8 = 96 cm².
2. Find the area of a triangle with base 9 m and perpendicular height 4 m.
Answer: ½ × 9 × 4 = 18 m².
3. A parallelogram has base 14 cm and perpendicular height 6 cm. Find its area.
Answer: 14 × 6 = 84 cm².
4. A trapezium has parallel sides 8 cm and 14 cm, separated by 5 cm. Find its area.
Answer: ½ × (8 + 14) × 5 = 55 cm².
5. Convert 2.5 m² to cm².
Answer: 2.5 × 10,000 = 25,000 cm².
Chapter summary
Area measures a two-dimensional region in square units. Select a formula that matches the shape, use perpendicular height, decompose complex regions and convert units before calculating. Estimate and check the units at the end.