1. Chapter at a Glance
How much paint covers a wall? How much land does a farm field occupy? How many tiles are needed for a floor? All these are questions about area — the amount of surface a flat shape covers. You already know the area of a rectangle and a square. In this chapter we use the clever idea of cutting and rearranging to find areas of parallelograms, triangles, trapeziums, rhombuses and any polygon.
- Units of area and conversions.
- Area of a parallelogram from a rectangle.
- Area of a triangle as half a parallelogram.
- Area of a trapezium.
- Area of a rhombus and any quadrilateral using diagonals.
- Area of irregular polygons by splitting them.
2. Units of Area
Area is measured in square units: cm², m², km², hectares.
| Conversion | Why |
|---|---|
| 1 cm² = 100 mm² | 10 mm × 10 mm |
| 1 m² = 10,000 cm² | 100 cm × 100 cm |
| 1 hectare = 10,000 m² | 100 m × 100 m |
| 1 km² = 100 hectares | 1000 m × 1000 m = 10⁶ m² |
Length units convert by 10, 100, 1000, but area units convert by the square: 1 m = 100 cm but 1 m² = 10,000 cm².
3. Area of a Parallelogram
Cut a right-angled triangle from one side of a parallelogram and slide it to the other side. The parallelogram becomes a rectangle with the same base and the same height.
Area of parallelogram = base × height (height is the perpendicular distance between the base and the opposite side — not the slanting side).
Any side can be the base, as long as we use the height drawn to that side: b₁ × h₁ = b₂ × h₂.
A parallelogram has sides 12 cm and 8 cm. The height on the 12 cm side is 6 cm. Find the area and the height on the 8 cm side.
Solution: Area = 12 × 6 = 72 cm². Height on 8 cm side = 72 ÷ 8 = 9 cm.
4. Area of a Triangle
Two copies of any triangle fit together to make a parallelogram with the same base and height. So the triangle is half of that parallelogram.
Area of triangle = ½ × base × height
- Right triangle: the two legs act as base and height → ½ × leg₁ × leg₂.
- For an obtuse triangle, the height may fall outside the triangle — extend the base to meet it.
- All triangles on the same base and between the same parallel lines have equal area.
The area of a triangle is 54 cm² and its base is 12 cm. Find its height.
Solution: ½ × 12 × h = 54 → 6h = 54 → h = 9 cm.
5. Area of a Trapezium
A trapezium has one pair of parallel sides (a and b) at distance h apart. Join two copies, one turned upside down, and you get a parallelogram of base (a + b) and height h. The trapezium is half of it.
Area of trapezium = ½ × (sum of parallel sides) × distance between them = ½(a + b)h
The parallel sides of a trapezium-shaped field are 30 m and 50 m, and they are 20 m apart. Find its area.
Solution: ½ × (30 + 50) × 20 = 800 m².
6. Rhombus and General Quadrilateral
The diagonals of a rhombus cut each other at right angles. Each diagonal splits the rhombus into two triangles, giving:
Area of rhombus = ½ × d₁ × d₂ (product of diagonals ÷ 2). This also works for a square and a kite.
Any quadrilateral: draw one diagonal d. Drop perpendiculars h₁ and h₂ from the other two vertices onto it. Then
A rhombus has diagonals 16 cm and 12 cm. Find its area and its side.
Solution: Area = ½ × 16 × 12 = 96 cm². Half-diagonals 8 and 6 meet at 90°, so side = √(64 + 36) = 10 cm.
7. Area of Irregular Polygons
Split the polygon into rectangles, triangles and trapeziums, find each area, and add. For a field survey, take one long diagonal as a baseline and measure perpendicular "offsets" to each corner.
An L-shaped room is made of a 6 m × 4 m rectangle joined to a 3 m × 2 m rectangle. Find the floor area.
Solution: 24 + 6 = 30 m².
A regular hexagon of side 6 cm is split by a long diagonal into two equal trapeziums with parallel sides 12 cm and 6 cm and height about 5.2 cm. Find its area.
Solution: Each trapezium: ½ × (12 + 6) × 5.2 = 46.8 cm². Hexagon ≈ 93.6 cm².
8. Solved Problems (NCERT Style)
A floor 8 m × 5 m is to be covered with square tiles of side 50 cm. How many tiles are needed? If each tile costs ₹40, find the total cost.
Solution: Floor = 40 m²; tile = 0.25 m². Tiles = 40 ÷ 0.25 = 160. Cost = 160 × 40 = ₹6,400.
A trapezium has area 180 cm², height 12 cm and one parallel side 18 cm. Find the other parallel side.
Solution: ½(18 + b) × 12 = 180 → 6(18 + b) = 180 → 18 + b = 30 → b = 12 cm.
9. Master Formula Table
| Shape | Area |
|---|---|
| Rectangle | l × b |
| Square | s² or ½ d² |
| Parallelogram | base × height |
| Triangle | ½ × base × height |
| Right triangle | ½ × leg₁ × leg₂ |
| Trapezium | ½ (a + b) h |
| Rhombus / kite | ½ d₁ d₂ |
| General quadrilateral | ½ d (h₁ + h₂) |
| 1 hectare | 10,000 m² |
10. Master Summary Table
| Shape | Comes from | Exam focus |
|---|---|---|
| Parallelogram | Rectangle (cut & slide) | Two heights, two bases |
| Triangle | Half a parallelogram | Find missing height |
| Trapezium | Half of a parallelogram of base (a + b) | Fields, missing parallel side |
| Rhombus | Two triangles on a diagonal | Side from diagonals |
| Polygon | Split into known shapes | Survey offsets, L-shapes |
Chart — areas of shapes with base 10 cm and height 6 cm:
11. Mind Map
- cm², m², hectare
- Square the factor
- Parallelogram
- Triangle = half
- Trapezium ½(a+b)h
- Rhombus ½d₁d₂
- Quadrilateral ½d(h₁+h₂)
12. Tips and Tricks
- Every formula here grows from "rectangle = l × b" by cutting and rearranging — remember the picture, not just the formula.
- Convert all lengths to the same unit before multiplying.
- For a rhombus, half-diagonals and a side form a right triangle — use Baudhāyana–Pythagoras.
- Using the slanting side instead of the perpendicular height.
- Forgetting the ½ in triangle, trapezium and rhombus formulas.
- Converting m² to cm² by multiplying by 100 instead of 10,000.
13. Important One-liners
- Area of a parallelogram = base × perpendicular height.
- A triangle has half the area of a parallelogram with the same base and height.
- Trapezium area = half the sum of the parallel sides × the distance between them.
- Rhombus area = half the product of its diagonals.
- 1 hectare = 10,000 m².
- Triangles on the same base and between the same parallels have equal area.
14. Exam-pattern Questions with Detailed Answers
Find the area of a parallelogram with base 15 cm and height 8 cm.
Show answer
15 × 8 = 120 cm².
Answer: 120 cm²
Find the area of a triangle with base 14 cm and height 9 cm.
Show answer
½ × 14 × 9 = 63 cm².
Answer: 63 cm²
How many cm² are in 3 m²?
- 300
- 3,000
- 30,000
- 3,00,000
Show answer
1 m² = 10,000 cm², so 3 m² = 30,000 cm².
Answer: (C)
Find the area of a rhombus with diagonals 24 cm and 10 cm, and its perimeter.
Show answer
Area = ½ × 24 × 10 = 120 cm². Side = √(12² + 5²) = 13 cm; perimeter = 52 cm.
Answer: 120 cm², perimeter 52 cm
A trapezium has parallel sides 9 cm and 15 cm and area 96 cm². Find the distance between the parallel sides.
Show answer
½ × 24 × h = 96 → 12h = 96 → h = 8 cm.
Answer: 8 cm
In a quadrilateral, a diagonal is 20 cm and the perpendiculars to it from the other two vertices are 7 cm and 5 cm. Find its area.
Show answer
½ × 20 × (7 + 5) = 120 cm².
Answer: 120 cm²
A right triangle has legs 9 cm and 12 cm. Find its area and the height on the hypotenuse.
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Area = ½ × 9 × 12 = 54 cm². Hypotenuse = 15 cm. Height on it: ½ × 15 × h = 54 → h = 7.2 cm.
Answer: 54 cm², 7.2 cm
A field of area 2.5 hectares is rectangular with length 250 m. Find its breadth.
Show answer
2.5 ha = 25,000 m². Breadth = 25000 ÷ 250 = 100 m.
Answer: 100 m
A parallelogram has adjacent sides 10 cm and 16 cm. The distance between the longer sides is 5 cm. Find the distance between the shorter sides.
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Area = 16 × 5 = 80 cm². Distance between shorter sides = 80 ÷ 10 = 8 cm.
Answer: 8 cm
A rhombus has area 240 cm² and one diagonal 16 cm. Find the other diagonal and its side.
Show answer
½ × 16 × d = 240 → d = 30 cm. Side = √(8² + 15²) = 17 cm.
Answer: 30 cm; side 17 cm
The parallel sides of a trapezium are in the ratio 3 : 5, the height is 14 cm and the area is 448 cm². Find the parallel sides.
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Let sides 3k and 5k. ½ × 8k × 14 = 448 → 56k = 448 → k = 8. Sides: 24 cm and 40 cm.
Answer: 24 cm and 40 cm
A 2 m wide path runs all around the inside of a rectangular garden 30 m × 20 m. Find the area of the path and the cost of paving it at ₹150 per m².
Show answer
Inner rectangle: 26 m × 16 m = 416 m². Garden = 600 m². Path = 184 m². Cost = 184 × 150 = ₹27,600.
Answer: 184 m²; ₹27,600
15. Quick Revision Summary
- Area units scale by the square: 1 m² = 10,000 cm²; 1 ha = 10,000 m².
- Parallelogram = b × h; triangle = ½ b h.
- Trapezium = ½(a + b)h.
- Rhombus = ½ d₁d₂; any quadrilateral = ½ d(h₁ + h₂).
- Irregular shapes: split, find each area, add.