Chapter 6 : Perimeter and Area
6.1 Perimeter
The perimeter of a closed flat figure is the total length you cover when you go once around its boundary. For a polygon (a closed figure made of straight line segments), the perimeter is simply the sum of the lengths of all its sides.
Perimeter of a Rectangle
Take a rectangle ABCD with length 12 cm and breadth 8 cm. Opposite sides of a rectangle are always equal, so AB = CD and AD = BC.
= 2 × AB + 2 × BC
= 2 × (12 cm + 8 cm)
= 2 × 20 cm = 40 cm
Perimeter of a Square
All four sides of a square are equal. So instead of adding all sides, we multiply one side by 4. For a square photo frame of side 1 m, the tape needed all around = 1 + 1 + 1 + 1 = 4 m.
Perimeter of a Triangle
For a triangle with sides 4 cm, 5 cm and 7 cm, perimeter = 4 + 5 + 7 = 16 cm.
Perimeter of a Regular Polygon
A regular polygon is a closed figure whose sides are all equal and whose angles are all equal (for example an equilateral triangle or a regular pentagon).
So for an equilateral triangle, perimeter = 3 × length of a side. A square and an equilateral triangle are similar in one way: in both, every side has the same length, so the perimeter is just one side length multiplied by the number of sides.
6.1 Figure it Out (Page 132)
1. Find the missing terms.
b. Perimeter of square = 20 cm → side = 20 ÷ 4 = 5 cm
c. Perimeter = 12 m, length = 3 m → 12 = 2 × (3 + b) → 3 + b = 6 → breadth = 3 m
2. A rectangle of sides 5 cm and 3 cm is made from a wire. The same wire is bent into a square. Find the side of the square.
3. A triangle has perimeter 55 cm with two sides 20 cm and 14 cm. Find the third side.
4. Find the cost of fencing a rectangular park of length 150 m and breadth 120 m, if fencing costs ₹40 per metre.
5. A string 36 cm long is used to form each shape. Find the side length for:
b. Equilateral triangle: 36 ÷ 3 = 12 cm
c. Regular hexagon: 36 ÷ 6 = 6 cm
6. A farmer's rectangular field is 230 m long and 160 m wide. He fences it with 3 rounds of rope. Find the total rope length.
6.1 Figure it Out — Running Tracks (Page 133)
Akshi runs on the outer track (70 m × 40 m) and Toshi runs on the inner track (60 m × 30 m).
| Runner | One round (Perimeter) | Rounds | Total distance |
|---|---|---|---|
| Akshi (outer) | 2 × (70 + 40) = 220 m | 5 | 1100 m |
| Toshi (inner) | 2 × (60 + 30) = 180 m | 7 | 1260 m |
1. Total distance Akshi covered in 5 rounds.
2. Total distance Toshi covered in 7 rounds. Who ran longer?
3. Marking positions (each round = full lap).
Toshi (round = 180 m): After 250 m → 1 round done, 70 m more. After 500 m → 2 rounds (360 m) and 140 m more. After 1000 m → 1000 ÷ 180 = 5 full rounds with 100 m left over (mark Z).
6.1 Deep Dive — Common Finish Line (Page 134)
Inner track = square of side 100 m (round = 400 m). Outer track = square of side 150 m (round = 600 m). The race is 350 m and both must finish at the same flag.
Where should each runner start?
Outer runner (B): Working back 350 m from the flag = 125 + 150 + 75 = 350 m.
Both start so they cover exactly 350 m to reach the common flag.
6.1 Straight and Diagonal Units (Page 134–135)
On dot paper, red lines (straight, one unit) and blue lines (diagonal) have different lengths. A diagonal is always longer than a straight side, so we write perimeters separately as straight units (s) and diagonal units (d).
Write the perimeters of the letter-shaped figures in straight (s) and diagonal (d) units.
6.1 Split and Rejoin (Page 136)
A 6 cm × 4 cm chit is cut into two equal pieces and rejoined in different ways. When the pieces join, the shared edges are hidden inside, so the perimeter changes even though the area stays the same.
| Arrangement | Perimeter |
|---|---|
| a. Long strip (6 cm + 6 cm × 2 cm) | 28 cm |
| b. L-shape (2 cm step) | 28 cm |
| c. T / plus shape | 28 cm |
| d. Offset stack (3 cm shift) | 28 cm |
Imp To make a figure with perimeter 22 cm, join the two 6 × 2 pieces along their long (6 cm) edges to rebuild the original 6 × 4 rectangle: perimeter = 2 × (6 + 4) = 20 cm; joining along the short edges gives the strip. Overlapping the longest edges gives the smallest perimeter.
6.2 Area
The area is the amount of flat region enclosed by a closed figure. Area is measured in square units.
Area of a square = side × side
Worked Idea — Floor and Carpet
A floor 5 m × 4 m has area 20 sq m. A square carpet of side 3 m has area 9 sq m. Floor not covered = 20 − 9 = 11 sq m.
Worked Idea — Flower Beds
Land 12 m × 10 m = 120 sq m. Four square beds of side 4 m each = 4 × 16 = 64 sq m. Remaining land = 120 − 64 = 56 sq m.
6.2 Figure it Out (Page 138)
1. A rectangular garden 25 m long has area 300 sq m. Find its width.
2. Cost of tiling a plot 500 m × 200 m at ₹8 per hundred sq m.
3. A coconut grove 100 m × 50 m, each tree needs 25 sq m. Maximum trees?
4. Split each figure into rectangles and find the area.
6.2 Estimating Area Using Squares
To find the area of an odd shape, trace it onto graph paper and use these rules:
- One full square = 1 sq unit.
- Ignore parts smaller than half a square.
- If more than half a square is inside, count it as 1 sq unit.
- If exactly half is covered, count it as ½ sq unit.
Find the area of the letter-shaped grid figures (Page 140).
Why Squares? (Let's Explore)
Circles cannot be packed tightly — they leave gaps, so counting them gives different totals (42 or 44 for the same box). Squares fit together perfectly with no gaps or overlaps, which makes them the best shape for measuring area.
Rectangles With Area 24 sq units (Let's Explore)
| Length × Width | Perimeter |
|---|---|
| 24 × 1 | 50 units (greatest) |
| 12 × 2 | 28 units |
| 8 × 3 | 22 units |
| 6 × 4 | 20 units (least) |
6.3 Area of a Triangle
Draw a rectangle and cut it along a diagonal. You get two triangles that overlap exactly, so they have equal areas. Each triangle is half the rectangle.
This works even for a triangle whose top vertex is not above a corner (triangle ABE). Dropping a straight line from the top splits it into two right triangles, each of which is half of a small rectangle. Adding them gives half of the whole rectangle.
6.3 Figure it Out (Page 144)
1. Find the areas of the figures by splitting them into rectangles and triangles.
6.3 Making It 'More' or 'Less' (Page 145)
Using 9 unit squares (area always 9 sq units), the shape can be arranged to give different perimeters.
| Question | Answer |
|---|---|
| 1. Smallest perimeter | 12 units (a 3 × 3 square) |
| 2. Largest perimeter | 20 units (a 9 × 1 strip) |
| 3. Figure with perimeter 18 | An L / staircase arrangement of the 9 squares gives 18 units |
| 4. More than one shape? | Yes for 18 and 20 units, but only one shape (the 3 × 3 square) gives 12 units |
Imp When a new square is attached: if it touches the figure along one edge the perimeter increases by 2; along two edges it stays the same; along three edges it decreases by 2. So a square can be placed to make the perimeter increase, decrease, or stay the same.
6.3 House Plans — Charan (Page 146)
The plot is a rectangle, 30 ft tall. Using the known rooms we find the missing sizes.
| Room | Dimensions | Area |
|---|---|---|
| Master Bedroom | 15 ft × 15 ft | 225 sq ft |
| Small Bedroom | 15 ft × 12 ft | 180 sq ft |
| Toilet | 5 ft × 10 ft | 50 sq ft |
| Kitchen | 15 ft × 12 ft | 180 sq ft |
| Utility | 15 ft × 3 ft | 45 sq ft |
| Hall | 20 ft × 12 ft | 240 sq ft |
| Parking | 15 ft × 3 ft | 45 sq ft |
| Garden | 20 ft × 3 ft | 60 sq ft |
Whole house = 35 ft × 30 ft = 1050 sq ft.
6.3 House Plans — Sharan (Page 147)
The plot is 42 ft wide.
| Room | Dimensions | Area |
|---|---|---|
| Master Bedroom | 12 ft × 15 ft | 180 sq ft |
| Small Bedroom | 12 ft × 10 ft | 120 sq ft |
| Toilet | 5 ft × 10 ft | 50 sq ft |
| Kitchen | 18 ft × 10 ft | 180 sq ft |
| Utility | 7 ft × 10 ft | 70 sq ft |
| Hall | 23 ft × 15 ft | 345 sq ft |
| Entrance | 7 ft × 15 ft | 105 sq ft |
Whole house = 42 ft × 25 ft = 1050 sq ft.
Area of Charan's house = Area of Sharan's house = 1050 sq ft (equal areas).
Perimeter of Charan's house = 130 ft; Perimeter of Sharan's house = 134 ft.
So Sharan's house has the greater perimeter even though both have the same area — same area can go with different perimeters.
6.3 Area Maze Puzzles (Page 148)
Find the missing area or side by comparing rows and columns that share the same width or height.
| Puzzle | Answer |
|---|---|
| a. | 30 sq cm |
| b. | 9 sq cm |
| c. | 16 sq cm |
| d. | 5 cm |
6.3 Figure it Out (Page 149)
1. Give the dimensions of a rectangle whose area equals the sum of the areas of 5 m × 10 m and 2 m × 7 m.
2. A rectangular garden 50 m long has area 1000 sq m. Find its width.
3. A room floor is 5 m × 4 m; a 3 m square carpet is laid. Find the uncarpeted area.
4. Four beds of 2 m × 1 m are dug at the four corners of a 15 m × 12 m garden. Area left for the lawn?
5. Shape A has area 18 sq units and a longer perimeter than Shape B (area 20 sq units). Draw two such shapes.
Shape B (area 20) can be 5 × 4 (P = 18), 10 × 2 (P = 24) or 20 × 1 (P = 42).
Since Shape A must have the longer perimeter, pick for example Shape A = 2 × 9 (P = 22) and Shape B = 5 × 4 (P = 18), then draw them.
6. Draw a rectangular border 1 cm from top and bottom and 1.5 cm from left and right on a page. Find its perimeter.
7. Draw a 12 × 8 rectangle, then draw an inner rectangle (not touching it) with exactly half the area.
8. A square is folded in half and cut into two rectangles. Which statement is always true?
Square perimeter = 8; each rectangle perimeter = 6; two together = 12 = 1½ × 8.
So option (c) is correct — the perimeters of both rectangles added together are always 1½ times the perimeter of the square.
Imp Formulas for Exams
- Perimeter of a rectangle = 2 × (length + breadth)
- Perimeter of a square = 4 × side
- Perimeter of a regular polygon = number of sides × one side
- Area of a rectangle = length × width
- Area of a square = side × side
- Area of a triangle = half the area of a rectangle with the same base and height
- Same area can have different perimeters, and same perimeter can have different areas.