11- Mensuration

Exercise 11.1

Question 1- A square and a rectangular field with measurements as given in the figure have the same perimeter. Which field has a larger area?

Answer - Perimeter of square = 4 (Side of the square) = 4 (60 m) = 240 m

Perimeter of rectangle = 2 (Length + Breadth)

= 2 (80 m + Breadth)

= 160 m + 2 × Breadth

It is given that the perimeter of the square and the rectangle are the same.

160 m + 2 × Breadth = 240 m

Breadth of the rectangle =  = 40 m

Area of square = (Side)² = (60 m)² = 3600 m²

Area of rectangle = Length × Breadth = (80 × 40) m² = 3200 m²

Thus, the area of the square field is larger than the area of the rectangular field.

Question 2- Mrs. Kaushik has a square plot with the measurement as shown in the following figure. She wants to construct a house in the middle of the plot. A garden is developed around the house. Find the total cost of developing a garden around the house at the rate of Rs 55 per m².

Answer -  Area of the square plot = (25 m)² = 625 m²

Area of the house = (15 m) × (20 m) =300 m²

Area of the remaining portion = Area of square plot − Area of the house

= 625 m² − 300 m² = 325 m²

The cost of developing the garden around the house is Rs 55 per m².

Total cost of developing the garden of area 325 m² = Rs (55 × 325)

= Rs 17,875

Question 3- The shape of a garden is rectangular in the middle and semi circular at the ends as shown in the diagram. Find the area and the perimeter of the garden [Length of rectangle is 20 − (3.5 + 3.5) metres]

Answer - Length of the rectangle = [20 − (3.5 + 3.5)] metres = 13 m

Circumference of 1 semi-circular part = πr 

Circumference of both semi-circular parts = (2 × 11) m = 22 m

Perimeter of the garden = AB + Length of both semi-circular regions BC and

DA + CD

= 13 m + 22 m + 13 m = 48 m­

Area of the garden = Area of rectangle + 2 × Area of two semi-circular regions

Question 4- A flooring tile has the shape of a parallelogram whose base is 24 cm and the corresponding height is 10 cm. How many such tiles are required to cover a floor of area 1080 m²? (If required you can split the tiles in whatever way you want to fill up the corners).

Answer - Area of parallelogram = Base × Height

Hence, area of one tile = 24 cm × 10 cm = 240 cm²

Required number of tiles = 

= 45000 tiles

Thus, 45000 tiles are required to cover a floor of area 1080 m².

Question 5- An ant is moving around a few food pieces of different shapes scattered on the floor. For which food − piece would the ant have to take a longer round? Remember, circumference of a circle can be obtained by using the expression c = 2πr, where r is the radius of the circle.

Answer - (a)Radius (r) of semi-circular part = 

Perimeter of the given figure = 2.8 cm + πr

(b)Radius (r) of semi-circular part = 

Perimeter of the given figure = 1.5 cm + 2.8 cm + 1.5 cm +π (1.4 cm)

(c)Radius (r) of semi-circular part = 

Perimeter of the figure(c) = 2 cm + πr + 2 cm

Thus, the ant will have to take a longer round for the food-piece (b), because the perimeter of the figure given in alternative (b) is the greatest among all.

Exercise -11.2

Question 1-  The shape of the top surface of a table is a trapezium. Find its area if its parallel sides are 1 m and 1.2 m and perpendicular distance between them is 0.8 m.

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Answer- Area of trapezium = (Sum of parallel sides) × (Distances between parallel sides)

Question 2- The area of a trapezium is 34 cm² and the length of one of the parallel sides is 10 cm and its height is 4 cm. Find the length of the other parallel side.

Answer - It is given that,area of trapezium = 34 cm² and height = 4 cm

Let the length of one parallel side be a. We know that,

Area of trapezium = (Sum of parallel sides) × (Distances between parallel sides)

Thus, the length of the other parallel side is 7 cm.

Question 3-  Length of the fence of a trapezium shaped field ABCD is 120 m. If BC = 48 m, CD = 17 m and AD = 40 m, find the area of this field. Side AB is perpendicular to the parallel sides AD and BC.

Answer -  Length of the fence of trapezium ABCD = AB + BC + CD + DA

120 m = AB + 48 m + 17 m + 40 m

AB = 120 m − 105 m = 15 m

Area of the field ABCD 

Question 4-  The diagonal of a quadrilateral shaped field is 24 m and the perpendiculars dropped on it from the remaining opposite vertices are 8 m and 13 m. Find the area of the field.

Answer -  It is given that,

Length of the diagonal, d = 24 m

Length of the perpendiculars, h₁ and h₂, from the opposite vertices to the diagonal are h₁ = 8 m and h₂ = 13 m

Area of the quadrilateral 

Thus, the area of the field is 252 m².

Question 5- The diagonals of a rhombus are 7.5 cm and 12 cm. Find its area.

Answer -  Area of rhombus = (Product of its diagonals)

Therefore, area of the given rhombus

= 

= 45 cm²

^{Question 6-} Find the area of a rhombus whose side is 6 cm and whose altitude is 4 cm. If one of its diagonals is 8 cm long, find the length of the other diagonal.

Answer - Let the length of the other diagonal of the rhombus be x.

A rhombus is a special case of a parallelogram.

The area of a parallelogram is given by the product of its base and height.

Thus, area of the given rhombus = Base × Height = 6 cm × 4 cm = 24 cm²

Also, area of rhombus = (Product of its diagonals)

Thus, the length of the other diagonal of the rhombus is 6 cm.

Question 7- The floor of a building consists of 3000 tiles which are rhombus shaped and each of its diagonals are 45 cm and 30 cm in length. Find the total cost of polishing the floor, if the cost per m² is Rs 4

Answer -  Area of rhombus = (Product of its diagonals)

Area of each tile

= 675 cm²

Area of 3000 tiles = (675 × 3000) cm² = 2025000 cm² = 202.5 m²

The cost of polishing is Rs 4 per m².

Cost of polishing 202.5 m² area = Rs (4 × 202.5) = Rs 810

Thus, the cost of polishing the floor is Rs 810.

Question 8- Mohan wants to buy a trapezium shaped field. Its side along the river is parallel to and twice the side along the road. It the area of this field is 10500 m² and the perpendicular distance between the two parallel sides is 100 m, find the length of the side along the river.

Answer -  Let the length of the field along the road be l m. Hence, the length of the field along the river will be 2l m.

Area of trapezium = (Sum of parallel sides) (Distance between the parallel sides)

Thus, length of the field along the river = (2 × 70) m = 140 m

Question 9- Top surface of a raised platform is in the shape of a regular octagon as shown in the figure. Find the area of the octagonal surface.

Answer -  

Side of regular octagon = 5 cm

Area of trapezium ABCH = Area of trapezium DEFG

Area of rectangle HGDC = 11 × 5 = 55 m²

Area of octagon = Area of trapezium ABCH + Area of trapezium DEFG

+ Area of rectangle HGDC

= 32 m² + 32 m² + 55 m² = 119 m²

^{Question 10-} There is a pentagonal shaped park as shown in the figure.

For finding its area Jyoti and Kavita divided it in two different ways.

Find the area of this park using both ways. Can you suggest some other way of finding its area?

Question 11-  Diagram of the adjacent picture frame has outer dimensions = 24 cm × 28 cm and inner dimensions 16 cm × 20 cm. Find the area of each section of the frame, if the width of each section is same.

Answer -  

Given that, the width of each section is same. Therefore,

IB = BJ = CK = CL = DM = DN = AO = AP

IL = IB + BC + CL

28 = IB + 20 + CL

IB + CL = 28 cm − 20 cm = 8 cm

IB = CL = 4 cm

Hence, IB = BJ = CK = CL = DM = DN = AO = AP = 4 cm

Area of section BEFC = Area of section DGHA

Area of section ABEH = Area of section CDGF

Exercise -11.3

Question 1- There are two cuboidal boxes as shown in the adjoining figure. Which box requires the lesser amount of material to make?

Answer -  We know that,

Total surface area of the cuboid = 2 (lh + bh + lb)

Total surface area of the cube = 6 (l)²

Total surface area of cuboid (a) = [2{(60) (40) + (40) (50) + (50) (60)}] cm²

= [2(2400 + 2000 + 3000)] cm²

= (2 × 7400) cm²

= 14800 cm²

Total surface area of cube (b) = 6 (50 cm)² = 15000 cm²

Thus, the cuboidal box (a) will require lesser amount of material.

Question 2- A suitcase with measures 80 cm × 48 cm × 24 cm is to be covered with a tarpaulin cloth. How many metres of tarpaulin of width 96 cm is required to cover 100 such suitcases?

Answer - Total surface area of suitcase = 2[(80) (48) + (48) (24) + (24) (80)]

= 2[3840 + 1152 + 1920]

= 13824 cm²

Total surface area of 100 suitcases = (13824 × 100) cm² = 1382400 cm²

Required tarpaulin = Length × Breadth

1382400 cm² = Length × 96 cm

Length = = 14400 cm = 144 m

Thus, 144 m of tarpaulin is required to cover 100 suitcases.

Question 3- Find the side of a cube whose surface area is 600 cm².

Answer - Given that, surface area of cube = 600 cm²

Let the length of each side of cube be l.

Surface area of cube = 6 (Side)²

600 cm² = 6l²

l²= 100 cm²

l = 10 cm

Thus, the side of the cube is 10 cm.

Question 4-  Rukhsar painted the outside of the cabinet of measure 1 m × 2 m × 1.5 m. How much surface area did she cover if she painted all except the bottom of the cabinet?

Answer - Length (l) of the cabinet = 2 m

Breadth (b) of the cabinet = 1 m

Height (h) of the cabinet = 1.5 m

Area of the cabinet that was painted = 2h (l + b) + lb

= [2 × 1.5 × (2 + 1) + (2) (1)] m²

= [3(3) + 2] m²

= (9 + 2) m²

= 11 m²

^{Question 5-} Daniel is painting the walls and ceiling of a cuboidal hall with length, breadth and height of 15 m, 10 m and 7 m respectively. From each can of paint 100 m² of area is painted. How many cans of paint will she need to paint the room?

Answer -  Given that,

Length (l) = 15 m, breadth (b) = 10 m, height (h) = 7 m

Area of the hall to be painted = Area of the wall + Area of the ceiling

= 2h (l + b) + lb

= [2(7) (15 + 10) + 15 ×10] m²

= [14(25) + 150] m²

= 500 m²

It is given that 100 m² area can be painted from each can.

Number of cans required to paint an area of 500 m²

=

Hence, 5 cans are required to paint the walls and the ceiling of the cuboidal hall.

Question 6- Describe how the two figures at the right are alike and how they are different. Which box has larger lateral surface area?

Answer -  Similarity between both the figures is that both have the same heights.

The difference between the two figures is that one is a cylinder and the other is a cube.

Lateral surface area of the cube = 4l² = 4 (7 cm)² = 196 cm²

Lateral surface area of the cylinder = 2πrh  cm² = 154 cm²

Hence, the cube has larger lateral surface area.

Question 7- A closed cylindrical tank of radius 7 m and height 3 m is made from a sheet of metal. How much sheet of metal is required?

Answer -  Total surface area of cylinder = 2πr (r + h)

 m²

= 440 m²

Thus, 440 m² sheet of metal is required.

Question 8- The lateral surface area of a hollow cylinder is 4224 cm². It is cut along its height and formed a rectangular sheet of width 33 cm. Find the perimeter of rectangular sheet?

Answer -  A hollow cylinder is cut along its height to form a rectangular sheet.

Area of cylinder = Area of rectangular sheet

4224 cm² = 33 cm × Length

Thus, the length of the rectangular sheet is 128 cm.

Perimeter of the rectangular sheet = 2 (Length + Width)

= [2 (128 + 33)] cm

= (2 × 161) cm

= 322 cm

Question 9- A road roller takes 750 complete revolutions to move once over to level a road. Find the area of the road if the diameter of a road roller is 84 cm and length is 1 m.

Answer - In one revolution, the roller will cover an area equal to its lateral surface area.

Thus, in 1 revolution, area of the road covered = 2πrh

In 750 revolutions, area of the road covered

=

= 1980 m²

^{Question 10-} A company packages its milk powder in cylindrical container whose base has a diameter of 14 cm and height 20 cm. Company places a label around the surface of the container (as shown in the figure). If the label is placed 2 cm from top and bottom, what is the area of the label.

Answer - Height of the label = 20 cm − 2 cm − 2 cm = 16 cm

Radius of the label 

Label is in the form of a cylinder having its radius and height as 7 cm and 16 cm.

Area of the label = 2π (Radius) (Height)

Exercise -11.4

Question 1- Given a cylindrical tank, in which situation will you find surface area and in which situation volume.

(a) To find how much it can hold

(b) Number of cement bags required to plaster it

(c) To find the number of smaller tanks that can be filled with water from it.

Answer - (a) In this situation, we will find the volume.

(b) In this situation, we will find the surface area.

(c) In this situation, we will find the volume.

Question 2- Diameter of cylinder A is 7 cm, and the height is 14 cm. Diameter of cylinder B is 14 cm and height is 7 cm. Without doing any calculations can you suggest whose volume is greater? Verify it by finding the volume of both the cylinders. Check whether the cylinder with greater volume also has greater surface area?

Answer -  The heights and diameters of these cylinders A and B are interchanged.

We know that,

Volume of cylinder 

If measures of r and h are same, then the cylinder with greater radius will have greater area.

Radius of cylinder A = cm

Radius of cylinder B = cm = 7 cm

As the radius of cylinder B is greater, therefore, the volume of cylinder B will be greater.

Let us verify it by calculating the volume of both the cylinders.

Volume of cylinder A

Volume of cylinder B

Volume of cylinder B is greater.

Surface area of cylinder A 

Surface area of cylinder B 

Thus, the surface area of cylinder B is also greater than the surface area of cylinder A.

Question 3-  Find the height of a cuboid whose base area is 180 cm² and volume is 900 cm³?

Answer -  Base area of the cuboid = Length × Breadth = 180 cm²

Volume of cuboid = Length × Breadth × Height

900 cm³ = 180 cm² × Height

Thus, the height of the cuboid is 5 cm.

Question 4- A cuboid is of dimensions 60 cm × 54 cm × 30 cm. How many small cubes with side 6 cm can be placed in the given cuboid?

Answer -  Volume of cuboid = 60 cm × 54 cm × 30 cm = 97200 cm³

Side of the cube = 6 cm

Volume of the cube = (6)³ cm³ = 216 cm³

Required number of cubes = 

Thus, 450 cubes can be placed in the given cuboid.

Question 5- Find the height of the cylinder whose volume is 1.54 m³ and diameter of the base is 140 cm?

Answer -  Diameter of the base = 140 cm

Radius (r) of the base 

Volume of cylinder 

Thus, the height of the cylinder is 1 m.

Question 6-  A milk tank is in the form of cylinder whose radius is 1.5 m and length is 7 m. Find the quantity of milk in litres that can be stored in the tank?

Answer - Radius of cylinder = 1.5 m

Length of cylinder = 7 m

Volume of cylinder 

1m³ = 1000 L

Required quantity = (49.5 × 1000) L = 49500 L

Therefore, 49500 L of milk can be stored in the tank.

Question 7- If each edge of a cube is doubled,

(i) how many times will its surface area increase?

(ii) how many times will its volume increase?

Answer - (i) Let initially the edge of the cube be l.

Initial surface area = 6l²

If each edge of the cube is doubled, then it becomes 2l.

New surface area = 6(2l)² = 24l² = 4 × 6l²

Clearly, the surface area will be increased by 4 times.

(ii) Initial volume of the cube = l³

When each edge of the cube is doubled, it becomes 2l.

New volume = (2l)³ = 8l³ = 8 × l³

Clearly, the volume of the cube will be increased by 8 times.