NCERT Solutions for Class 8 Maths Chapter 11 Mensuration

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.

Answer :

Area of top trapezium surface of the table=(\frac{1}2\times (sum\; of \;parallel \;sides)\times (distances \;between\; them)\)

\(=\frac{1}2\times(1+1.2)\times0.8 m^2\)

\(=\frac{1}2\times 2.2 \times 0.8 m^2\)

\(=(1.1\times 0.8)=0.88m^2\)

So, the area of top trapezium surface of the table=\(0.88m^2\)

2.The area of a trapezium is \(34 cm^2\) 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 :

Given, area of trapezium is \(34cm^2\).
Let the required side be x cm.

We know that, area of trapezium=\(\frac{1}2\times (sum\; of \;parallel \;sides)\times (distances \;between\; them)\)

\(34=\frac{1}2\times(10+x)\times 4\)

\(34=2(10+x)\)

\(10+x=17\)

\(x=7\)

Hence, the required side is 7cm.

3.Length of the fence of a trapezium A 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 :

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

AB + BC + CD + DA = 120 m .

AB = 120 m – (48 m + 17 m + 40 m) = 120 – 105 m = 15 m

So the area of trapezium fence ABCD=\(\frac{1}2\times (sum\; of \;parallel \;sides)\times (distances \;between\; them)\)

\(=\frac{1}2\times (BC+AD)\times AB\)

\(=\frac{1}2\times (48+40)\times 15\)

\(=\frac{1}2\times 88\times 15\)

\(=44\times 15=660m^2\)

So, the required area=\(660 m^2\)

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 :

As per the figure we have area of field=area of ∆ABD+area of ∆BCD

\(=\frac{1}2\times base \times height+\frac{1}2\times base \times height\)

\(=\frac{1}2\times 24 \times 13+\frac{1}2\times 24\times 8\)

\(=12\times 13+12\times 8\)

\(=12\times(13+8)\)

\(=12\times 21\)

\(=252m^2\)

Hence the required area of the given field=\(252m^2\)

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

Answer :

Given,d1 = 7.5 cm, d2 = 12 cm Area of a rhombus =\( \frac { 1 }{ 2 } x (product of diagonals)\)

\(=\frac{1}2\times (d1\times d2)\)

\(=\frac{1}2\times (7.5 \times 12)\)

\(=7.5\times 6\)

\(=45 cm^2\)

So the area of rhombus=45 \(cm^2\)

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.

Answer :

Given, Side = 5 cm, Altitude = 4.8 cm and length of one diagonal(d1) = 8 cm
We know, area of the rhombus = \(Side \times Altitude = 5 \times 4.8 = 24 cm^2\)

Area of the rhombus =\(\frac { 1 }{ 2 } \times d1 \times d2\)

\(\Rightarrow 24 = \frac { 1 }{ 2 } \times 8 \times d2\)

\(\Rightarrow 24 = 4\times d2\)

\(\Rightarrow d2 = 6 cm\)

Hence, the length of other diagonal = 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 \(m^2\) is ₹ 4.

Answer :

Given, number of tiles = 3000 and the length of the two diagonals of a tile = 45 cm and 30 cm
We have area of one rhombus shaped tile =\(\frac { 1 }{ 2 }\times d1 \times d2\)

\(=\frac { 1 }{ 2 }\times 45 \times 30\)

\(= 45 \times 15\)

\(= 675 cm^2\)

So, area of one tile=675\(cm^2\)

Therefore, Area covered by 3000 tiles = \(3000 \times 675 cm^2 = 2025000 cm^2 = 202.5 m^2\)

So, the cost of polishing the floor = \(202.5 \times 4 = ₹ 810\quad\)[∵ cost per \(m^2\) is ₹ 4. ]

Hence, the required cost = ₹ 810.

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 \(m^2\) and the perpendicular distance between the two parallel sides is 100m, find the length of the side along the river.

Answer :

Given, area of trapezium field area=10500\(m^2\), Altitude of the field(h)=100m.
Let, the side of the trapezium near road be x cm and the opposite parallel side = 2x m

We know, area of trapezium = \(\frac { 1 }{ 2 }\times (a + b) \times h\)

\(\Rightarrow 10500 = \frac { 1 }{ 2 }\times (2x + x) \times 100\)

\(\Rightarrow 2 \times10500 = 3x \times 100\)

\(\Rightarrow 21000 = 300x\)

\(\Rightarrow x = 70 m\)

So, the other side = \(2x = 2 \times 70 = 140 m\)

Hence, the required length of the side along the river= 140 m.

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 :

We can observe, Area of the octagonal surface= area of trapezium ABCH + area of rectangle HCDG + area of trapezium GDEF
Finding area of different parts:

Area of trapezium ABCH = Area of trapezium GDEF

\(=\frac { 1 }{ 2 }\times (a + b) \times h\)

\(=\frac { 1 }{ 2 }\times (11 + 5) \times 4\)

\(=\frac { 1 }{ 2 }\times 16 \times 4\)

\(= 32 m^2\)

Area of rectangle HCDG =\( l \times b = 11 m \times 5 m = 55 m^2\)

Area of the octagonal surface = \(32 m^2 + 55 m^2 + 32 m^2 = 119 m^2\)

Hence, the required area = \(119 m^2\).

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?

Answer :

(i) From Jyoti’s diagram:


We can observe that, Area of the pentagonal shape = Area of trapezium ABCD + Area of trapezium ADEF

\(= 2 \times Area \;of\; trapezium \;ABCD\)

\(= 2 \times \frac { 1 }{ 2 } \times (a + b) \times h\)

\(= (15 + 30) \times 7.5\)

\(= 45 \times 7.5\)

\(= 337.5 m^2\)

So, the area of the pentagonal shape=337.5\(m^2\)

(ii) From Kavita’s diagram:



We can see that, area of the pentagonal shape = Area of triangle ABE + Area of square BCDE

\(=\frac { 1 }{ 2 }\times b \times h + 15 \times 15\)

\(=\frac { 1 }{ 2 }\times 15 \times 15 + 225\)

\(= 112.5 + 225= 337.5 m^2\)

So, the area of the pentagonal shape =\(337.5 m^2\)

Yes, there is other way to find the area which is as follows:



Required area ABCDEA=Area of rect. ABPQ-2\(\times\)area of triangle CPD

\(=(15\times 30)-(2\times \frac{1}2\times \frac{15}2)\)

\(=(450-\frac{225}2)=\frac{(900-225)}2m^2\)

\(=\frac{675}2m^2=337.5m^2\)

11.Diagram of the picture frame has outer dimensions = \(24 cm \times 28 cm\) and inner dimensions \(16 cm \times 20 cm\). Find the area of each section of the frame, if the width of each section is the same.

Answer :

We have:
\(h_1=\frac{1}2(28-20)=\frac{1}2\times 8=4cm\)

\(h_2=\frac{1}2(24-16)=\frac{1}2\times 8=4cm\)

We also know, area of trapezium A=\(\frac{1}2\times(a+b)\times h_1\)

\(=\frac{1}2\times(24+16)\times4\)

\(=\frac{1}2\times 48\times 4=80cm^2\)

so, the area of trapezium A=area of trapezium C=80\(cm^2\)

Now, area of trapezium B=Area of trapezium D

\(=\frac{1}2\times(28+20)\times4\)

\(=\frac{1}2\times 48\times 4=96cm^2\)

Hence, we have areas of all the parts as:

Part A=\(80cm^2\)

Part B=\(96cm^2\)

Part C=\(80cm^2\)

Part D=\(96cm^2\)

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

Answer :

We have, the total surface area of first box=\( 2(lb + bh + lh)\)
\(= 2 (60 \times 40 + 40 \times 50 + 60 \times 50) cm^2\)

\(= 200 (24+20+30) cm^2\)

\(= 200 \times 74 cm^2=14800 cm^2\)

Also, the total surface area of second box= \(6 (Edge)^2\)

\( = 6 \times 50 \times 50 cm^2= 15000 cm^2\)

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

2.A suitcase of measures \(80 cm \times 48 cm \times 24 cm\) 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 :

Measurement of the suitcase =\( 80 cm \times 48 cm \times 24 cm\)

We can say that l = 80 cm, b = 48 cm and h = 24 cm So the total surface area of the suitcase = \(2[lb + bh + hl]\)

\(= 2 (80 \times 48 + 48 \times 24 + 80 \times 24) cm^2\)

\(= 2(3840 + 1152 + 1920) cm^2\)

\(= 2 \times 6912 cm^2 =13824 cm^2\)

We have, Area of tarpaulin = Area of 100 suitcase

Area of Area of 100 suitcase=\(100 \times 13824\)

Also, we have the Area of tarpaulin = \(length \times breadth = l \times 96 = 96l cm^2\)

Equating both we have:

\(96l =100 \times 13824\)

\(\Rightarrow l = 100 \times 144 = 14400 cm = 144 m\)

Hence, the required length of the cloth = 144 m.

3.Find the side of a cube whose surface area is \(600 cm^2\)?

Answer :

Given, the surface area of cube is \(600 cm^2\)
Total surface area of a cube =\( 6l^2\)

\(\Rightarrow 6l^2 = 600\)

\(\Rightarrow l^2 = 100\)

\(\Rightarrow l = \sqrt{100} = 10 cm\)

Hence, the required length of side = 10 cm.

4.Rukhsar painted the outside of the cabinet of measure \(1 m \times 2 m \times 1.5 m\). How much surface area did she cover if she painted all except the bottom of the cabinet.

Answer :

Given, l = 2 m, b = 1.5 m, h = 1 m
Area of the surface to be painted = Total surface area of box – Area of base of box

\(= 2 (lb + bh + hl) – lb\)

\(= 2(2 \times 1.5 + 1.5 \times 1 + 1 \times 2) – 2 \times 1\)

\(= 2(3 + 1.5 + 2) – 2\)

\(= 2(6.5) – 2\)

\(= 13 – 2= 11 m^2\)

Hence, the required area = 11 \(m^2\).

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 \(m^2\) of area is painted.How many cans of paint will she need to paint the room?

Answer :

Given, l = 15 m, b = 10 m and h = 7 m
Area of the paint in one can = \(100 m^2\)

So,Surface area of a cuboidal hall without bottom = Total surface area – Area of base

\(= 2 (lb + bh + hl) – lb\)

\(= 2 (15 \times 10 + 10 \times 7 + 7 \times 15) – 15 \times 10\)

\(= 2(150 + 70 + 105) – 150\)

\(= 2 (325) – 150\)

\(= 650 – 150= 500 m^2\)

Therefore, the number of cans required =\(\frac { 500 }{ 100 }= 5 cans\).

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

Answer :

The two figures given differ in their shapes.One of them is cylinder and the other is cube.

Similarity: Both the given figures have the same height.

Given, Cylinder: d = 1 cm, h = 7 cm

Cube: Length of each side a = 7 cm

So, Lateral surface of cylinder = 2\(\pi \)rh

\(= 2 \times \frac { 22 }{ 7 }\times \frac { 7 }{ 2 } \times 7 = 154 cm^2\)

Also, the Lateral surface of the cube =\( 4l^2 = 4 \times (7)^2 = 4 \times 49 = 196cm^2\)

So, the cube has the larger lateral surface =\( 196 cm^2\).

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 :

Given, r = 7 m and h = 3 m.
Sheet required to make a closed cylinder= Total surface area of the cylinder

\(= 2\pi r(r + h)\) sq. units

\(=2\times \frac { 22 }{ 7 } \times 7\times (3+7)\)

\(=2\times \frac { 22 }{ 7 } \times 7\times 10 \)

\(=440 m^2\)

So, the required metal sheet=\(440m^2\)

8.The lateral surface area of a hollow cylinder is 4224 \(cm^2\). It is cut along its height and formed a rectangular sheet of width 33 cm. Find the perimeter of the rectangular sheet.

Answer :

Given, The lateral surface area of a hollow cylinder is 4224 \(cm^2\) and the width of rectangular sheet =33cm
We know that the circumference of the base of the cylinder is equal to the length of the sheet and the height of the cylinder is equal to the breadth of the sheet when a hollow cylinder is cut and a rectangular sheet is formed. Therefore, the lateral surface area of the cylinder is equal to the area of the rectangular sheet.

Let the length of the sheet be l.

∴ we have ,area of the sheet = Lateral surface area of cylinder

\(\Rightarrow l \times 33 = 4224\)

\(\Rightarrow l=\frac{4224}{33}=128cm\)

Thus, the length of the rectangular sheet=128cm\)

Now, perimeter of rectangular sheet= 2 (l + b)

\( = 2 (128 + 33) cm\)

\(= 2 \times 161 cm=322 cm\)

Hence, the perimeter of the rectangular sheet=322cm.

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 :

Given,diameter of the road roller=84cm and length=1m.Also revolution=750

We have radius of roller r=\(\frac{84}2=42cm=0.42m\)

So, curved surface area of the roller=\(2\pi rh\)

\(=2\times \frac{22}7 \times 0.42 \times 1\)

\(=22.64 m^2\)

∴area levelled by the roller in 750 revolutions:

\(\qquad=(750\times 2.64)=1980m^2\)

Hence, the area of the road levelled=\(1980m^2\)

10.A company packages its milk powder in cylindrical containers 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.

Answer :

Given, Diameter of the cylindrical container=14cm, height=20cm, label placed=2cm from top to bottom.

So the height covered by the label=20-(2+2)=16cm

Radius of the container=\(\frac{1}2\times d=\frac{1}2\times 14=7cmm\)

Surface area of the cylindrical shaped label =\( 2\pi rh\)

\(= 2 \times\frac { 22 }{ 7 } \times 7 \times 16\)

\(= 704 cm^2\)

Hence, the area of the label on container =\(704 cm^2.\)

1.Given a cylindrical tank, in which situation will you find the 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 :

In this situation we can find:
(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 it by finding the volume of both the cylinders. Check whether the cylinder with greater volume also has greater surface area?

Answer :

Cylinder B will have a greater volume.
Verification:

Volume of cylinder A=\(\pi r^2h\)

\(=\frac{22}7\times \frac{7}2\times\frac{7}2\times 14\)

\(=539cm^3\)

Volume of cylinder B=\(\pi r^2h\)

\(=\frac{22}7\times7\times7\times 7\)

\(=22\times49=1078cm^3\)

So, we can see that cylinder B has more volume.Hence the assumption was correct.

Now comparing the surface areas of both the cylinders:

Total surface area of cylinder A=\(2\pi r(r+h)\)

\(=2\times\frac{22}7\times\frac{7}2(\frac{7}2+14)\)

\(=2\times\frac{22}7\times\frac{7}2\times \frac{35}2\)

\(=385cm^2\)

Total surface area of cylinder B=\(2\pi r(r+h)\)

\(=2\times\frac{22}7\times7(7+7)\)

\(=2\times\frac{22}7\times7\times 14\)

\(=616cm^2\)

Hence we can observe that cylinder B has more surface area.

3.Find the height of a cuboid whose base area is 180 \(cm^2\) and volume is 900 \(cm^3\)?

Answer :

Given,Volume of the cuboid =900 \(cm^3\) and Base area=180 \(cm^2\) As we know, \( (Area of the base) \times Height = 900 cm^3\)

\(\Rightarrow 180 \times Height = 900\)

\(\Rightarrow Height = \frac { 900 }{ 180 } = 5\)

Hence, the required height of the cuboid= 5 cm.

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

Answer :

Given, Volume of the cuboid \((l \times b \times h )= 60 cm \times 54 cm \times 30 cm = 97200 cm^3\) and sie of cube=6cm
So, we have Volume of the cube= \((Side)^3= (6)^3 = 216 cm^3\)

We know, Number of the cubes from the cuboid=\(\frac{Volume\;of\;the\;cuboid}{volume\;of\;cube}\)

\(=\frac{97200}{216}=450\)

Hence, the required number of cubes=450

5.Find the height of the cylinder whose volume is 1.54 \(m^3\\) and diameter of the base is 140 cm?

Answer :

Given, V = 1.54 \(m^3\), d = 140 cm = 1.40 m[∵1m=100cm]
We have,Volume of the cylinder = \(\pi r^2h\)

\(\Rightarrow 1.54=\frac{22}7\times{1.4}2\times\frac{1.4}2\times h\quad\)[(∵\(r=\frac{d}2=\frac{1.4}2\)]

\(\Rightarrow h=\frac{1.5\times2\times2\times7}{1.4\times22\times1.4}=1m\)

Hence, the height of the cylinder =1m.

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 liters that can be stored in the tank?
,img src="https://trustudies-app-public.s3.ap-south-1.amazonaws.com/11.4(q6).png">

Answer :

Given, r = 1.5 m, h = 7 m
We have, Volume of the milk tank =\(\pi r^2h\)

\(=\frac { 22 }{ 7 }\times 1.5 \times 1.5 \times 7\)

\(= 22 \times 2.25= 49.50 m^3\)

∴ Volume of milk in litres = \(49.50 \times 1000 L\quad\) [∵ 1 \(m^3\) = 1000 litres]

\(= 49500 L\)

Hence, the required volume of milk = 49500 L.

7.If each edge of a cube is doubled,
(i) how many times will it be surface area increase?

(ii) how many times will its volume increase?

Answer :

Let the edge of the cube = x cm
As per the question, edge is doubled so the new edge = 2x cm

(i)We have original surface area =\( 6x^2 cm^2\)

And, the new surface area = \(6(2x)^2 = 6 \times 4\times2 = 24\times2\)

∴ Ratio = \(6x^2 : 24x^2 = 1 : 4\)

Hence, the new surface area will be four times the original surface area.

(ii)We have, original volume of the cube = \(x^3 cm^3\)

And, the new volume of the cube = \((2x)^3 = 8x^3 cm^3\)

∴Ratio =\( x^3 : 8x^3 = 1 : 8\)

Hence, the new volume will be eight times the original volume.

8.Water is pouring into a cuboidal reservoir at the rate of 60 liters per minute. If the volume of reservoir is 108 \(m^3\), find the number of hours it will take to fill the reservoir.

Answer :

Given,Volume of the reservoir = 108 \(m^3\) = 108000 L [∵1 \(m^3\) = 1000 L]

Water flowing into the reservoir per minute = 60 L

Time taken to fill the reservoir=\(\frac{Volume\;of\;reservoir}{Rate of flowing water}\)

\(=\frac{10800}{60}minutes=1800\;minutes\; or \;\frac{11800}{60}=30\; hours\)

Hence, the required hour to fill the reservoir = 30 hours.s