11 . Mensuration

Chapter 11 . Mensuration

11. Mensuration

Exercise 11.1

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

Solution: For square field : Here ,

The perimeter of the square field

For rectangular field : Here ,

The perimeter of the rectangular field

A/Q ,

(a) The area of the square field

(b) The area of the rectangular field

The square field has a larger area , i.e., figure (a)

2. Mrs. Kaushik has a square plot with the measurement as shown in the 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

Solution: For square plot : Here,

Area of the square plot

For rectangular plot : Here , and

Area of the rectangular plot

Area of the garden around the house

The total cost of developing a garden around the house

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 this garden [Length of rectangle is 20 – (3.5 + 3.5) metres].

Solution: Here ,

, and

Area of the garden = Area of two semi circular part + Area of rectangular part

The perimeter of the garden Circumference of two semi-circle + perimeter of rectangular – 2 × diameter

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

? (If required you can split the tiles in whatever way you want to fill up the corners).

Solution: Here , and

The area of the tile

The number of tiles are required to cover a floor

[ and ]

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.

Solution: (a) Here, Diameter , Radius

The circumference of the semi circle

(b) Here, , ,

The perimeter of the given figure

The ant have to take a longer round for figure (b) .

Exercise 11.2

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.

Solution: Here, , and

We know that , the area of the trapezium

2. The area of a trapezium is 34

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.

Solution: let be the length of the other parallel side.

Here, ,

A/Q , The area of the trapezium

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

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.

Solution: Given , BC = 48 m, CD = 17 m and AD = 40 m

The length of the fence of a trapezium shaped field ABCD = 120 m

Here, , and

Area of the trapezium

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.

Solution: Here, , and

We know that , The area of the quadrilateral field

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

Solution: Here , and

We know that ,

The area of the rhombus

6. Find the area of a rhombus whose side is 5 cm and whose altitude is 4.8 cm. If one of its diagonals is 8 cm long, find the length of the other diagonal.

Solution: Here , , and

The area of a rhombus

Let be the length of the other diagonal.

A/Q ,

Therefore , the length of the other diagonal is 6 cm .

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

is Rs 4.

Solution: Here, ,

The area of the rhombus shape tiles

The total cost of polishing the floor

8. Mohan wants to buy a trapezium shaped field. Its side along the river is parallel to and twice the side along the road. If the area of this field is 10500

and the perpendicular distance
between the two parallel sides is 100 m, find the length of the side along the river.

Solution: Here , ,

Let a be the length of the side along the road and be the length of the side along the river .

We know that ,

The area of the trapezium shaped field

Therefore, the length of the side along the river

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.

Solution: Here, , and

The area of the octagonal surface

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?

Solution: For Joyti’s diagram :

Here , ,and

Area of the pentagonal shaped park

For Kavita diagram : Here, and

Area of the pentagonal shaped park area of the square + Area of the triangle

Yes , we can also find the area of the pentagonal shape .

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.

Solution: For 1st section :

Here , , and

Area of the 1nd section

For 2nd section : Here , , and

Area of the 2nd section

Area of 1st section = Area of 3rd section

Area of 2nd section = Area of 4th section

Exercise 11.3

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

Solution: For figure (a) :

Here, , and

The surface area of cuboidal box

For figure (b) : Here ,

The surface area of cube

Therefore, the figure (a) (i.e., first figure) is the lesser amount of material required

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?

Solution: Here, ,and

The surface area of the cuboidal suitcase

The surface area of 100 suitcase

The length of the tatpaulin

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

Solution: let be the side of a cube .

A/Q,

Therefore, the side of a cube is 10 cm .

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.

Solution: Here, ,,

The surface area of the cabinet for paint

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

of area is painted. How many cans of paint will she need to paintthe room?

Solution: Here, , and

The surface area of the room

The number of the cans

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

Solution: Both the figures(cylinder and cube) are same height . So, the two figures at the right are alike .

Bothe the figure are different shape .

For cylinder :

Here, , and

The curve surface area of cylinder

For cube : Here ,

The surface area of cube

The cube has larger lateral surface area

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 ?

Solution: Here, ,

The total surface area of the cylindrical tank

Required the sheet of metal is 440 .

8. The lateral surface area of a hollow cylinder is 4224

. It is cut along its height and formed a rectangular sheet of width 33 cm. Find the perimeter of rectangular sheet?

Solution: Since , the hollow cylinder cut along its height .

The circumference of the cylindrical part

So,

A/Q ,

Here , and

The perimeter of rectangular sheet

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.

Solution: Here , Diameter

Radius

and length

The curve surface area of roller

The curve surface area of 750 complete revolution

Therefore, area of the roads is .

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.

Solution:

Here, ,

and

The curve surface area of cylindrical container

Therefore, the area of the label is .

Exercise 11.4

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.

Solution: (a) In this situation , the volume will be held .

(b) In this situation , the surface area will be held .

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?

Solution:
The volume of cylinder B is greater. Because the diameter of cylinder B is greater .

For cylinder A :

Here, , and

The volume of the cylinder A

For cylinder B :

Here, , and

The volume of the cylinder B

The surface area of cylinder :

For cylinder A :

Here, , and

The total curve surface area of cylinder A

For cylinder B :

Here, , and

The total curve surface area of the cylinder B

The surface area of cylinder B is greater .

3. Find the height of a cuboid whose base area is 180

and volume is 900

Solution : let be the height of a cuboid .

A/Q ,

Therefore, the height of the cuboid is 5 cm .

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?

Solution: For cuboid :

Here, , and

The volume of the cuboid

For cube : Here,

The volume of the cube

The number of small cube

5. Find the height of the cylinder whose volume is 1.54

and diameter of the base is 140 cm ?

Solution: let be the height of the cylinder .

Here, ,

and

A/Q,

Therefore, the height of the cylinder is 1 m .

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?

Solution: Here, and

The volume of the cylindrical tank

[ Note : ]

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?

Solution: (i) We know that , the surface area of cube

If the edge of a cube is doubled () , then

The surface area of cube

The surface area is increased 4 times .

(ii) We know that , the volume of cube

If the edge of a cube is doubled ( ) , then

The surface area of cube

The volume of the cube is increased 8 times .

8. Water is pouring into a cubiodal reservoir at the rate of 60 litres per minute. If the volume of reservoir is 108

, find the number of hours it will take to fill the reservoir.

Solution: The volume of the cubiodal reservoir

The water is pouring into a cuboidal reservoir

The number of hours will take to fill the reservoir