1. Chapter at a Glance
When a farmer puts a fence around a field, the length of the fence is the perimeter of the field. When a family covers a floor with tiles, the amount of floor covered is its area. Perimeter measures the boundary; area measures the surface inside the boundary. This chapter teaches how to find both for rectangles, squares, triangles and other shapes, and shows that shapes with the same perimeter can have different areas — and the other way round.
- Perimeter of rectangles, squares, triangles and regular polygons.
- Area as the number of unit squares that cover a region.
- Area of a rectangle and a square; estimating area on squared paper.
- Area of a triangle as half of a rectangle.
- Comparing shapes with equal perimeter or equal area.
- Real-life uses: fencing, tiling, painting and running tracks.
2. Perimeter
The perimeter of a closed figure is the total length of its boundary. For a figure made of straight sides, add the lengths of all sides. Perimeter is a length, so it is measured in mm, cm, m or km.
Perimeter of a rectangle
A rectangle has two lengths (l) and two breadths (b):
Perimeter of a square
A square has 4 equal sides of length a:
Perimeter of a triangle and regular polygons
- Triangle with sides a, b, c: perimeter = a + b + c.
- Equilateral triangle with side a: perimeter = 3 × a.
- Regular polygon with n equal sides of length a: perimeter = n × a (regular pentagon = 5a, regular hexagon = 6a).
A rectangular park is 80 m long and 45 m wide. How much wire is needed to fence it with 3 rounds of wire?
Solution: Perimeter = 2 × (80 + 45) = 2 × 125 = 250 m. For 3 rounds: 3 × 250 = 750 m of wire.
The perimeter of a square is 64 cm. Find its side.
Solution: 4 × side = 64, so side = 64 ÷ 4 = 16 cm.
Perimeter of a running track
Athletes in the inner lane run a shorter distance than those in outer lanes. That is why in races the outer-lane runners start a little ahead (a staggered start) — so that everyone runs the same distance.
3. Area
The area of a closed figure is the amount of surface it covers. We measure area by counting how many unit squares fit inside it. A square of side 1 cm has area 1 square centimetre (1 cm² or 1 sq cm). Larger areas use m² and km².
Area on squared paper
To estimate the area of an irregular shape on 1 cm squared paper:
- Count the full squares inside the shape — each is 1 cm².
- Count the squares that are about half inside — each is ½ cm².
- Count squares that are more than half inside as 1 cm²; ignore those less than half inside.
- Add them up to get the approximate area.
Area of a rectangle and a square
A rectangle 5 cm long and 3 cm wide holds 3 rows of 5 unit squares = 15 squares. In general:
A room floor is 6 m by 4 m. Square tiles of side 50 cm are to be laid. How many tiles are needed?
Solution: Area of floor = 6 × 4 = 24 m². Area of one tile = 0.5 × 0.5 = 0.25 m². Number of tiles = 24 ÷ 0.25 = 96. (Or: 600 cm ÷ 50 = 12 tiles along the length, 400 ÷ 50 = 8 along the width, 12 × 8 = 96.)
4. Area of a Triangle
Cut a rectangle along its diagonal — you get two equal triangles. So the area of each right triangle is half the rectangle. In fact, any triangle drawn inside a rectangle with the same base and the same height fills exactly half of it (the two leftover pieces outside the triangle can be fitted together to make another copy).
The height is the perpendicular distance from the opposite corner to the base.
Find the area of a triangle with base 12 cm and height 7 cm.
Solution: Area = ½ × 12 × 7 = 42 cm².
5. Same Perimeter, Different Area (and vice versa)
Perimeter and area are different measures. Changing the shape can change one without changing the other.
| Rectangle (l × b) | Perimeter | Area |
|---|---|---|
| 11 × 1 | 24 cm | 11 cm² |
| 10 × 2 | 24 cm | 20 cm² |
| 9 × 3 | 24 cm | 27 cm² |
| 8 × 4 | 24 cm | 32 cm² |
| 7 × 5 | 24 cm | 35 cm² |
| 6 × 6 (square) | 24 cm | 36 cm² |
All these rectangles have perimeter 24 cm, but the area increases as the shape becomes closer to a square. For a fixed perimeter, the square has the largest area.
Similarly, shapes with the same area (say 12 cm²) can have different perimeters: 12 × 1 gives 26 cm, 6 × 2 gives 16 cm, 4 × 3 gives 14 cm.
With 40 m of fencing, which rectangular garden encloses the most area?
Solution: l + b = 20. The area l × b is largest when l = b = 10 m (a square). Maximum area = 100 m².
6. Real-life Applications
- Fencing, framing, lace on a tablecloth → perimeter.
- Tiling a floor, painting a wall, carpeting, buying land → area.
- Cost problems: total cost = perimeter × rate per metre (for fencing) or area × rate per m² (for flooring).
Painting a wall 5 m long and 3 m high costs ₹40 per m². A door 2 m by 1 m is not to be painted. Find the cost.
Solution: Wall area = 5 × 3 = 15 m². Door = 2 × 1 = 2 m². Area to paint = 13 m². Cost = 13 × 40 = ₹520.
7. Master Formula Table
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle (l, b) | 2(l + b) | l × b |
| Square (side a) | 4a | a × a |
| Triangle (sides a, b, c; base, height) | a + b + c | ½ × base × height |
| Equilateral triangle (side a) | 3a | — |
| Regular polygon (n sides, side a) | n × a | — |
| Missing side of rectangle | b = P/2 − l | b = Area ÷ l |
| Side of square | a = P ÷ 4 | a × a = Area |
| Unit conversion | Value |
|---|---|
| 1 cm | 10 mm |
| 1 m | 100 cm |
| 1 km | 1000 m |
| 1 m² | 100 × 100 = 10,000 cm² |
| 1 cm² | 10 × 10 = 100 mm² |
8. Master Summary Table
| Idea | Perimeter | Area |
|---|---|---|
| Measures | Length of boundary | Surface covered |
| Units | cm, m, km | cm², m², km² |
| Everyday use | Fencing, borders, tracks | Tiles, paint, land |
| Fixed perimeter | — | Square gives maximum area |
| Fixed area | Long thin shapes give larger perimeter | — |
Chart: area of rectangles with perimeter 24 cm
9. Mind Map
- Rectangle 2(l + b)
- Square 4a
- Regular polygon n × a
- Unit squares
- Rectangle l × b
- Triangle ½ b h
- Same perimeter, different area
- Square has most area
- Fencing, tiling
- Painting costs
- Running tracks
10. Tips and Tricks
- Always convert all lengths to the same unit before calculating.
- For tiling, divide length by tile side and breadth by tile side, then multiply — this avoids decimals.
- Half the perimeter of a rectangle = l + b. Use this to find a missing side quickly.
- When subtracting doors and windows, find their total area first, then subtract once.
- Writing area in cm instead of cm².
- Using 4 × side for the area of a square (that is the perimeter).
- Forgetting the ½ in the area of a triangle.
- Thinking that a larger perimeter always means a larger area.
- Converting 1 m² as 100 cm² (it is 10,000 cm²).
11. Important One-liners
- Perimeter is the length of the boundary of a closed figure.
- Perimeter of a rectangle = 2 × (length + breadth).
- Perimeter of a square = 4 × side.
- Area is the measure of the surface enclosed by a figure.
- Area of a rectangle = length × breadth.
- Area of a square = side × side.
- Area of a triangle = ½ × base × height.
- A diagonal divides a rectangle into two triangles of equal area.
- Among rectangles with the same perimeter, the square has the greatest area.
- 1 m² = 10,000 cm².
- Outer-lane runners start ahead because outer lanes are longer.
12. Exam-pattern Questions with Detailed Answers
Find the perimeter of a rectangle of length 14 cm and breadth 9 cm.
- 23 cm
- 46 cm
- 126 cm
- 32 cm
Show answer
Perimeter = 2 × (14 + 9) = 2 × 23 = 46 cm.
Answer: (B) 46 cm
The area of a square is 81 m². Find its side and perimeter.
Show answer
side × side = 81, so side = 9 m. Perimeter = 4 × 9 = 36 m.
Answer: Side 9 m; perimeter 36 m
A regular hexagon has a perimeter of 54 cm. What is the length of each side?
Show answer
A hexagon has 6 equal sides: 54 ÷ 6 = 9 cm.
Answer: 9 cm
The perimeter of a rectangle is 60 m and its length is 18 m. Find its breadth and area.
Show answer
Half perimeter = l + b = 30, so b = 30 − 18 = 12 m.
Area = 18 × 12 = 216 m².
Answer: Breadth 12 m; area 216 m²
How many square tiles of side 20 cm are needed to cover a floor 4 m by 3 m?
Show answer
Along 400 cm: 400 ÷ 20 = 20 tiles. Along 300 cm: 300 ÷ 20 = 15 tiles.
Total = 20 × 15 = 300 tiles.
Answer: 300 tiles
Fencing costs ₹25 per metre. Find the cost of fencing a square plot of area 900 m².
Show answer
Side = 30 m (since 30 × 30 = 900). Perimeter = 4 × 30 = 120 m.
Cost = 120 × 25 = ₹3,000.
Answer: ₹3,000
A triangle has base 15 cm and area 60 cm². Find its height.
- 4 cm
- 8 cm
- 12 cm
- 16 cm
Show answer
½ × 15 × h = 60 → 15h = 120 → h = 8 cm.
Answer: (B) 8 cm
A square and a rectangle have the same perimeter of 36 cm. The rectangle is 12 cm long. Which has the greater area, and by how much?
Show answer
Square side = 36 ÷ 4 = 9 cm, area = 81 cm².
Rectangle breadth = 18 − 12 = 6 cm, area = 72 cm².
Answer: The square, by 9 cm²
A rectangular garden 30 m by 20 m has a path 2 m wide all around it outside. Find the area of the path.
Show answer
Outer rectangle = (30 + 4) × (20 + 4) = 34 × 24 = 816 m².
Garden = 30 × 20 = 600 m². Path = 816 − 600 = 216 m².
Answer: 216 m²
Using 24 square tiles of side 1 m, Meena wants a rectangular patio with the smallest possible perimeter. Which dimensions should she choose?
Show answer
Possible rectangles with area 24: 24 × 1 (P = 50), 12 × 2 (P = 28), 8 × 3 (P = 22), 6 × 4 (P = 20).
The shape closest to a square, 6 × 4, has the smallest perimeter.
Answer: 6 m × 4 m (perimeter 20 m)
The length of a rectangle is doubled and its breadth is halved. What happens to its area and perimeter? Try l = 8 cm, b = 6 cm.
Show answer
Original: area 48 cm², perimeter 28 cm. New: 16 × 3 → area 48 cm² (same), perimeter 2 × 19 = 38 cm (larger).
Area stays the same in general (2l × b/2 = lb), but the perimeter usually changes.
Answer: Area unchanged; perimeter becomes 38 cm here
A wall 6 m long and 4 m high has one window 1.5 m × 1 m and one door 2 m × 1 m. Painting costs ₹35 per m². Find the total cost.
Show answer
Wall = 24 m². Window = 1.5 m², door = 2 m². Area to paint = 24 − 3.5 = 20.5 m².
Cost = 20.5 × 35 = ₹717.50.
Answer: ₹717.50
13. Quick Revision Summary
- Perimeter = boundary length: rectangle 2(l + b), square 4a, triangle a + b + c, regular polygon n × a.
- Area = surface covered in square units: rectangle l × b, square a², triangle ½ × base × height.
- Estimate irregular areas by counting full and half squares on squared paper.
- Same perimeter can give different areas; for a fixed perimeter the square has the most area.
- Use consistent units; 1 m² = 10,000 cm².