Class 6 Mathematics · Chapter 6

6. Perimeter and Area

Answers to all questions on perimeter, area, tangram pieces, triangles on grid paper and house plans. CBSE / NCERT Class 6 Ganita Prakash.

6.1 Perimeter — Figure it Out

1Find the missing terms: (a) Perimeter of a rectangle = 14 cm, breadth = 2 cm, length = ? (b) Perimeter of a square = 20 cm, side = ? (c) Perimeter of a rectangle = 12 m, length = 3 m, breadth = ?

(a) Length + breadth = 14 ÷ 2 = 7 cm. Length = 7 − 2 = 5 cm

(b) Side = 20 ÷ 4 = 5 cm

2A rectangle of sides 5 cm and 3 cm is made from a wire. The wire is bent to form a square. What is the side of the square?

Length of wire = perimeter of rectangle = 2 × (5 + 3) = 16 cm

Side of square = 16 ÷ 4 = 4 cm

3A triangle has perimeter 55 cm and two sides 20 cm and 14 cm. Find the third side.

Third side = 55 − (20 + 14) = 55 − 34 = 21 cm

4Find the cost of fencing a rectangular park 150 m long and 120 m wide at ₹40 per metre.

Perimeter = 2 × (150 + 120) = 2 × 270 = 540 m

Cost = 540 × 40 = ₹21,600

5A string is 36 cm long. Find the length of each side if it forms (a) a square (b) an equilateral triangle (c) a regular hexagon.

(a) 36 ÷ 4 = 9 cm

(b) 36 ÷ 3 = 12 cm

6A field is 230 m long and 160 m wide. The farmer fences it with 3 rounds of rope. Find the total length of rope.

Perimeter = 2 × (230 + 160) = 2 × 390 = 780 m

Rope for 3 rounds = 3 × 780 = 2340 m

Matha Pachchi! — Figure it Out

1Find the total distance Akshi covered in 5 rounds.

One round = 2 × (70 + 40) = 220 m

5 rounds = 5 × 220 = 1100 m

2Find the total distance Toshi covered in 7 rounds. Who ran a longer distance?

One round = 2 × (60 + 30) = 180 m

7 rounds = 7 × 180 = 1260 m

Toshi ran a longer distance (1260 m > 1100 m).

3Mark the positions of Akshi (A, B, C) and Toshi (X, Y, Z).

Both start at the bottom-right corner and run along the bottom side first.

Point	Distance run	Position
A (Akshi)	250 m = 220 + 30	1 round + 30 m along the bottom side from the start
B (Akshi)	500 m = 440 + 60	2 rounds + 60 m along the bottom side (10 m before the corner)
C (Akshi)	1000 m = 880 + 120	4 full rounds + 120 m: 10 m along the top side from the top-left corner
X (Toshi)	250 m = 180 + 70	1 round + 70 m: 10 m up the left side
Y (Toshi)	500 m = 360 + 140	2 rounds + 140 m: 50 m along the top side from the top-left corner
Z (Toshi)	1000 m = 900 + 100	5 full rounds + 100 m: 10 m along the top side from the top-left corner

★Deep Dive: For a 350 m race with a common finishing line, mark the starting points A (inner track) and B (outer track).

Inner track (side 100 m): 350 = 100 + 100 + 100 + 50. So A is at the corner three sides and a half side before the flag (the bottom-right corner).

Outer track (side 150 m): 350 = 125 + 150 + 75. So B is on the top side, 125 m before the top-left corner (25 m from the top-right corner).

★Akshi says the perimeter of the triangle is 9 units. Toshi says it is more. What do you think?

Toshi is right. The slanting (diagonal) side of a square is longer than its straight side. So the perimeter is more than 9 units.

★Write the perimeters of the figures in terms of straight and diagonal units.

Figure 1: 8s + 2d

Figure 2: 4s + 6d

Figure 3: 12s + 6d

Figure 4: 18s + 6d

★What is a similarity between a square and an equilateral triangle?

Both have all sides equal and all angles equal. They are regular polygons.

Perimeter of a regular polygon = number of sides × length of one side.

★Split and rejoin: Find the perimeter of arrangements b, c and d. Arrange the pieces to get a perimeter of 22 cm.

Each piece is 6 cm × 2 cm, with perimeter 16 cm. When two pieces touch, the touching length is not counted on either piece.

(b) Touching length 2 cm: 32 − 2 × 2 = 28 cm

(d) Touching length 3 cm: 32 − 2 × 3 = 26 cm

22 cm: 32 − 22 = 10, so the touching length must be 5 cm. Place the pieces side by side along their long sides and slide one piece 1 cm.

6.2 Area

★Area of a square = ____ ; Area of a rectangle = ____

Area of a square = side × side

Area of a rectangle = length × width

6.2 — Figure it Out

1A rectangular garden 25 m long has area 300 sq m. Find its width.

Width = 300 ÷ 25 = 12 m

2Find the cost of tiling a plot 500 m long and 200 m wide at ₹8 per hundred sq m.

Area = 500 × 200 = 1,00,000 sq m

Number of hundreds = 1,00,000 ÷ 100 = 1000

Cost = 1000 × 8 = ₹8000

3A coconut grove is 100 m long and 50 m wide. Each tree needs 25 sq m. Find the maximum number of trees.

Area = 100 × 50 = 5000 sq m

Number of trees = 5000 ÷ 25 = 200

4Find the areas of the figures by splitting them into rectangles (in metres).

(a) Split into 4 rectangles: 3 × 3 = 9, 7 × 1 = 7, 5 × 2 = 10, 2 × 1 = 2

Area = 9 + 7 + 10 + 2 = 28 sq m

(b) Top strip 5 × 1 = 5, two legs 1 × 2 = 2 each

Area = 5 + 2 + 2 = 9 sq m

Tangram — Figure it Out

Let the area of a small triangle (Shape C) be 1 unit.

1Explore and figure out how many pieces have the same area.

A and B have the same area (4 units each).

C and E have the same area (1 unit each).

D, F and G have the same area (2 units each).

2How many times bigger is Shape D than Shape C? What is the relationship between Shapes C, D and E?

Shape D is 2 times Shape C.

Area of D = Area of C + Area of E.

3Which has more area: Shape D or F?

Both have the same area. Each can be covered exactly by the two small triangles C and E.

4Which has more area: Shape F or G?

Both have the same area (2 units each). Each is made of two small triangles.

5What is the area of Shape A compared to Shape G?

A = 4 units and G = 2 units. So A is twice as big as G.

6What is the area of the big square in terms of the area of Shape C?

4 + 4 + 1 + 1 + 2 + 2 + 2 = 16 times the area of Shape C.

7Arrange the 7 pieces to form a rectangle. What is its area in terms of Shape C?

The rectangle also has area 16 times Shape C, because it is made from the same 7 pieces.

8Are the perimeters of the square and the rectangle formed from the 7 pieces the same?

No. They have the same area, but different shapes. The rectangle has a longer perimeter than the square.

★Find the area of the figures on the dot grid.

Figure 1: 4 sq units

Figure 2: 9 sq units

Figure 3: 10 sq units

Figure 4: 11 sq units

Two half squares make one full square.

★Why is area generally measured using squares?

Squares fit together with no gaps and no overlaps.
All sides are equal, so a square looks the same in every direction.
Squares fit neatly along the length and width of a rectangle, so area = length × width.
Circles leave gaps, so they cannot measure area correctly.

★Make rectangles with whole-number sides and area 24 sq units. Which has the greatest perimeter? The least? What about 32 sq cm?

Rectangle (area 24)	Perimeter
1 × 24	50 units
2 × 12	28 units
3 × 8	22 units
4 × 6	20 units

(a) Greatest perimeter: 1 × 24. (b) Least perimeter: 4 × 6.

Prediction: The longest and thinnest rectangle has the greatest perimeter. The rectangle closest to a square has the least perimeter.

6.3 Area of a Triangle

★Cut a rectangle along a diagonal. Do the two triangles have the same area? What can you infer?

Yes. The two triangles overlap exactly.

Inference: A diagonal divides a rectangle into two equal triangles. Each triangle has half the area of the rectangle.

★Is the area of the blue rectangle more, less or the same as the yellow triangle? Why?

The same. The yellow triangle is made of the two halves of the blue rectangle (cut along its diagonal).

★Find the area of triangle BAD and triangle ABE.

Area of rectangle ABCD = 5 × 4 = 20 sq units

1. Area of triangle BAD = half of 20 = 10 sq units

2. Area of triangle ABE = half of 20 = 10 sq units

Conclusion: The area of a triangle with base AB and its third corner on the opposite side DC is half the area of the rectangle ABCD.

6.3 — Figure it Out

1Find the areas of the figures by dividing them into rectangles and triangles.

(a) Rectangle 4 × 5 = 20, two triangles of 2 each → 24 sq units

(b) Rectangle 4 × 5 = 20, triangles 6 and 4 → 30 sq units

(d) Rectangle 4 × 3 = 12, triangles 1 + 3 → 16 sq units

(e) Two triangles on a common base of 4: 4 + 8 → 12 sq units

Making it ‘More’ or ‘Less’

1Using 9 unit squares, what is the smallest perimeter possible?

12 units — a 3 × 3 square.

2What is the largest perimeter possible?

20 units — for example, a 1 × 9 strip, or any shape in which no 2 × 2 block is formed.

3Make a figure with a perimeter of 18 units.

Example: a 2 × 2 block of 4 squares with a straight tail of 5 squares attached to it.

4Can you make other shapes for each of the above perimeters?

For 12 units, only the 3 × 3 square is possible.

For 18 and 20 units, many different shapes are possible.

★A figure has perimeter 24 units. What happens when a new square is attached? Can the perimeter increase, decrease or stay the same?

In the position shown, the new square touches 2 sides of the figure. It adds 2 new sides and covers 2 old sides. So the perimeter stays the same (24 units).

(a) Increases: attach the square so that it touches only 1 side → perimeter becomes 26 units.

(b) Decreases: put the square in a gap where it touches 3 sides → perimeter becomes 22 units.

★Charan’s house plan: Find the missing measurements and the area of his house.

Room	Size	Area
Utility	15 ft × 3 ft	45 sq ft
Small bedroom	15 ft × 12 ft	180 sq ft
Hall	20 ft × 12 ft, plus a 5 ft × 5 ft passage below the toilet	240 + 25 = 265 sq ft
Garden	20 ft × 3 ft	60 sq ft
Parking	15 ft × 3 ft	45 sq ft

Length of the house = 15 + 5 + 15 = 35 ft; width = 30 ft

Area of the house = 35 × 30 = 1050 sq ft

★Sharan’s house plan: Find the missing measurements and the area. Compare with Charan’s house.

Room	Size	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

Area of the house = 42 × 25 = 1050 sq ft

Both houses have the same area (1050 sq ft).

Perimeter: Charan = 2 × (35 + 30) = 130 ft; Sharan = 2 × (42 + 25) = 134 ft. Sharan’s house has a longer perimeter.

★Area Maze Puzzles: Find the missing values.

(a) 26 is twice 13, so the right column is twice as wide as the left. ? = 2 × 15 = 30 sq cm

(b) Bottom rectangle: 10 ÷ 2 = 5 cm long. Tall rectangle width = 5 − 3 = 2 cm, height = 10 ÷ 2 = 5 cm. Pink part height = 5 − 2 = 3 cm. ? = 3 × 3 = 9 sq cm

(c) Middle rectangle: 42 ÷ 6 = 7 cm wide. Pink part width = 7 − 3 = 4 cm. Bottom width = 7 + 5 = 12 cm, height = 60 ÷ 12 = 5 cm. Pink height = 15 − 6 − 5 = 4 cm. ? = 4 × 4 = 16 sq cm

(d) The lower part of the big rectangle is as tall as the 18 sq cm rectangle. Try ? = 5 cm: top part = 5 × 4 = 20 sq cm and lower part = 18 sq cm. 20 + 18 = 38 ✓. So ? = 5 cm

Final Figure it Out

1Give the dimensions of a rectangle whose area is the sum of the areas of 5 m × 10 m and 2 m × 7 m.

Total area = 50 + 14 = 64 sq m

Rectangle: 8 m × 8 m (or 16 m × 4 m, or 32 m × 2 m)

2A rectangular garden 50 m long has area 1000 sq m. Find its width.

Width = 1000 ÷ 50 = 20 m

3A floor is 5 m × 4 m. A square carpet of side 3 m is laid. Find the area not carpeted.

Floor = 5 × 4 = 20 sq m; carpet = 3 × 3 = 9 sq m

Not carpeted = 20 − 9 = 11 sq m

4Four flower beds 2 m × 1 m are dug at the corners of a 15 m × 12 m garden. How much area is left for the lawn?

Garden = 15 × 12 = 180 sq m; flower beds = 4 × (2 × 1) = 8 sq m

Lawn = 180 − 8 = 172 sq m

5Shape A has area 18 sq units and Shape B has area 20 sq units. A has a longer perimeter. Draw such shapes.

Shape A: a 1 × 18 rectangle → perimeter = 2 × (1 + 18) = 38 units

Shape B: a 4 × 5 rectangle → perimeter = 2 × (4 + 5) = 18 units

A has less area but a longer perimeter.

6On a page of your book, draw a border 1 cm from the top and bottom and 1.5 cm from the left and right. What is its perimeter?

The border is 2 cm shorter in length and 3 cm shorter in width than the page.

So, perimeter of border = perimeter of page − 2 × (2 + 3) = perimeter of page − 10 cm.

Example: For a page 27 cm × 20 cm, the border is 25 cm × 17 cm. Perimeter = 2 × (25 + 17) = 84 cm.

7Draw a 12 × 8 rectangle. Draw another rectangle inside it, not touching it, with exactly half the area.

Area of outer rectangle = 12 × 8 = 96 sq units; half = 48 sq units.

Draw an 8 × 6 rectangle in the middle, leaving 2 units on the left and right and 1 unit at the top and bottom.

8A square is folded in half and cut into two rectangles. Which statement is always true?

Let the side of the square be s. Each rectangle is s × s2.

Perimeter of the square = 4s. Perimeter of each rectangle = 3s, so both together = 6s.

6s is 112 times 4s. So statement (c) is true.

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