Mensuration-NCERT Solutions

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

        

Sol. Let x be the breadth of the rectangle. It is given that the perimeter of a rectangle = perimeter of the square.

        ∴ 2(80 + x) = 4 × 60

        ⇒80 + x = 120

        ⇒x = 120 – 80 = 40

        i.e. breadth of the rectangle = 40 m

        Now, Area of the square = (60 × 60) m²

        = 3600 m²

        and the area of the rectangle = (40 × 80) m²

        = 3200 m²

        Hence, the square field has a larger area.

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. And the total cost of developing a garden around the house at the rate of Rs. 55 per m².

Sol. Area of the garden = Area of the outer square

        � Area of the inner rectangle

        = 25 × 25 m² � 20 × 15 m²

        (625 � 300) m² = 325 m²

        Cost of developing a garden @ Rs. 55 per sq. meter

        = Rs. (55 × 325) = Rs. 17875

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].

Sol. Total area of the garden

        = Area of the rectangular portion + The sum of the areas of the pair of semi circles

        

        Perimeter of the garden

= 2 × length of rectangular portion + circumference of circle

        

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?

Sol. Area of one tile = base × height

        = (24 × 10) cm²

        = 240 cm²

        Number of tiles required to cover the floor

        

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.

Sol. Let us mark points A, B, C and D in the given figures as shown. Let A be the point in each figure from where the ant start moving on the food pieces. She is to reach the initial point after moving around the boundary of each food piece.

        For food piece (a)

        Distance moved = Arc AB + BA

6.   The shape of the top surface of a table is trapezium. Find its area if its parallel sides are lm and 1.2 m and perpendicular distance between then is 0.8 m.

Sol. Area of top surface of a table

        = Area of the trapezium

        

7.   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 other parallel side.

Sol. Let the required side be x cm.

        Then, area of the trapezium

        

        But, the area of the trapezium = 34 cm² (given) ∴ 2 (10 + x) = 34

        ⇒10 + × = 17

        ⇒x = 17 – 10 = 7

        Hence, the outer side = 7 cm

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

Sol. Let ABCD be the given trapezium in which BC 48 m, CD = 17 m and AD = 40 m

        Through D, draw DL ⊥ BC.

        Now, BL = AD = 40 m

        and LC = BC – BL = (48 – 40) m = 8 m

        Applying pythagoras theorem in right ΔDLC, we have

        DL² = DC² – LC² = 17² – 8² = 289 – 64 = 225 ⇒ DL

        Now, area of the trapezium ABCD

        = (44 × 15) m² = 660 m².

9.   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.

Sol. Let ABCD be the given quadrilateral in which BE ⊥ AC and DF ⊥ AC.

        It is given that

        AC = 24 m, BE = 8 m and DF= 13 m.

        Now, area of quad. ABCD

        = area of ΔABC + area of ΔACD

        

        = (12 × 8 + 12 × 13) m²

        = (96 + 156) m² = 252 m²

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

Sol.

11.   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.

Sol. Let ABCD be a rhombus of side 6 cm and whose altitude DE = 4 cm. Also, one of its diagonals, BD = 8 cm.

        Area of the rhombus ABCD

        

        Hence, the other diagonal is 6 cm.

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

Sol. Area of the floor = 3000 � Area of one tile

        

13.   Mohan wants to b a trapezium shaped field. It side along the river is parallel to and twice the side along the road. If 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.

Sol. Let the parallel sides of the trapezium shaped field be x m and 2x m. Then, its area.

        

        ∴The length of the side along the river is 2 × 70, i.e., 140 metres.

14.   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.

Sol. Area of the octagonal surface ABCDEFGH = Area (trap. ARCH) + Area (rect. HCDG) + Area (trap. GDEF)

15.   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?

Sol. Taking Jyoti’s diagram:

        Area of the pentagonal shaped park

        = 2 × Area of trapezium ABEF

        

        Taking Kavita’s diagram:

        Area of the pentagonal shaped park

        

16.   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.

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

Sol. Total surface area of first box

        = 2(lb + bh + lh)

        = 2(60 × 40 + 40 × 50 + 60 × 50)cm²

        = 200(24 + 20 + 30) cm²

        = 200 × 74 cm² = 14800 cm²

        Total surface area of second box

        = 6 (Edge)2 = 6 × 50 × 50 cm²

        = 15000 cm²

        Since the total surface area of first box is less than that of the second, therefore the first box i.e., (a) requires the least amount of material to make.

2.   A suitcase of rneasures 80 cm x48 cm x24 cm is to be covered with a tarpaulin cloth. How many metres of tarpaulin of width 96 cm is required to cover 100 such suitacases?

Sol. Total surface area of suitcase

        = 2[(80) (48) + (48) (24) + (24) (80))

        = 2[3840 + 1152 + 1920]

        = 13824 cm²

        Total surface area of 100 suitcase

        = (13824 × 100) cm² =1382400 cm²

        Required tarpaulin = Length × Breadth

        1382400 cm² = Length x96 cm

        

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

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

Sol. Let a be the side of the cube having surface area 600 cm².

        ∴ 6a² = 600 ⇒ a2 = 100 ⇒ a = 10

        Hence, the side of the cube = 10 cm.

4.   Rukhsar painted the outside of the cabinet of measure 1 m x2m x1.5 m. How much surface area did she cover if she painted all except the bottom of the cabinate.

Sol. Here l = 2m, b = 1m and h = 1.5 m

        Area to be painted

        = 2bh + 2lh + lb

        (2 × 1 × 1.5 + 2 × 2 × 1.5 + 2 × 1)m²

        = (3 + 6 + 2) m² = 11m²

5.   Deniel 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?

Sol. Here l = 15m, b = 10 m and h = 7 m

        Area to be painted

        = 2bh + 2lh + lb

        = 2 × 10 × 7 + 2 × 15 × 7 + 15 × 10)m²

        = (140 + 210 150) m² = 500m²

        Since each can of paint covers 100 m², therefore number of cans required

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

Sol. Try yourself

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?

Sol. Here, r = 7m and h = 3 m.

        Sheet of metal required to make a closed cylinder = Total surface area of the cylinder. = (2πrh + 2πr²) sq. units.

        

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 perimetre of rectangular sheet.

Sol. A hollow cylind :r is cut alcng 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

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.

Sol. In one revolution, the roller will cover an area equal to its lateral surface area. Thus, in 1 revolution, area of the road covered = 27πrh

        

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 surface area of the label.

Sol. Since the company places a label around the surface of the cylindrical container of radius 7 cm and height 20 cm such that it is placed 2 cm from top and bottom.

        We have to find the curved surface of a cylinder of radius 7 cm and height (20 � 4) cm i.e., 16 cm.

        This curved surface area

        

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.

Sol. (a) volume (b) Surface area (c) Volume

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 by finding the volume of both the cylinders. Check whether the cylinder with greater volume also has greater surface area?

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

        We know that,

        Volume of cylinder = πr²h

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

        

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

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

        

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

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

Sol. Volume of the cuboid = 900 cm³

        ⇒ (Area of the base) × Height = 900 cm³ 180 × Height = 900

        

        Hence, the height of the cuboid is 5 cm.

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

Sol. Volume of cuboid = (60 × 54 × 30)cm³

        = 97200 cm³

        Volume of cube = (6 × 6 × 6) cm³

        = 216 cm³

        

5.   Find the height of the cylinder whose volume is 1.54 m³

Sol. Let the h be the height of cylinder whose radius,

        

        Hence, the height of cylinder is 1 metre.

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?

Sol. Quantity of milk that can be stored in the tank

        = Volume of the tank

        = πr²h, where r = 1.5 and h = 7 m

        

        = (49.5 × 1000) litres [∵ 1 m³ = 1000 Itr.]

        = 49500 litres.

7.   If each edge of a cube is doubled,

        (i) How many times Al its surface area increase?

        (ii) How many times will its volume increase?

Sol. Let x units be the edge of the cube. Then, its surface area = 6x² and its volume = x³.

        When its edge is doubled,

        (i) Its surf ace area = 6 (2x)² = 6 × 4x² = 24x²

                ⇒ The surface area of the new cube will be 4 times that of the original cube.

        (ii) Its volume = (2x)³ = 8x³

                ⇒ The volume of the new cube will be 8 times that of the original cube.

8.   Water is pouring into a cuboidal reservoir at the rate of 60 litres per minute. If the volume of reservoi: is 108 m³, find the number of hours it will take to fill the reservoir.

Sol. Volume of the reservoir = 108 m³

        = 108 × 1000 litres

        = 108000 litres

        Since water is pouring into reservoir @ 60 litres per minute.

        ∴ Time taken to fill the reservoir