mensuration - logical class

MATHEMATICS MENSURATION

1 VII - CLASS

§§ Mensuration :

Mensuration is the science of measurement; or, in a more limited sense, the science of numerical representation of geometrical magnitudes. It is a branch of mathematics whichdeals with formulae for calculating the numerical measurements of curved lengths,areas andvolumes, in terms of numerical data which determine these measurements.

The number, pi, which is the ratio of the circumference of a circle to its diameter, has a long story about its value. The notation for pi (the Greek letter ) was first used by William Jones in 1706. Once Euler adopted it, it became standard notation. The earliest known record of pi was written by an Egyptian scribe named Ahmes around 1650 BC, in the Rhind papyrus. In 250 BC, Archimedes was one of the first people to calculate pi by mathematical methods, when he worked it out to be between 3.1408 and 3.1429.

It was the computer that finally revolutionised pi calculations, when in 1949 the ENIAC Computer, with its 19,000 vacuum tubes and hundreds of thousands of resistors,found pi to 2037 places. Now, computers have calculated pi to billions of places,although it is not necessaryto know it to so many digits.

§§ Units of Measurement :

METRIC SYSTEM BRITISH SYSTEM

Length F.P.S Length

10 millimeters = 1 cm 12 inches = 1 foot

10 centimeters = 1 decimeter 3 ft. = 1 yard

10 Decimeters = 1 m 22 yds. = 1 chain

10 meters = 1 dm 10 chains = 1 furlong

10 decameters = 1 hm 8 furlongs = 1 mile

10 hectometers = 1 km (kilometers)

Note : 1 mile = 1760 yds = 5280 ft ; 1 nautical mile (knot) = 6080 ft

WEIGHT WEIGHT

10 mg (milligram) = 1 cg (centigram) 16 ounce (oz) = 1 lb (pound)

10 cg = 1 dg (decigram) 14 lbs = 1 stone

10 dg = 1 gm (gram) 2 stones = 1 quarter

10 gm = 1 dgm (decagram) 4 quarters = 1 cwt (hundred weight)

10 dgm = 1 hg (hectogram) 20 cwt = 1 ton

100 kg = 1 hg (hectogram)

Note : 1000 kg = 1 metric tonne ; 1 cwt = 112 lbs, 1 ton = 2240 lbs

CAPACITY (VOLUME) CAPACITY

10 ml (milliliters) = 1 (centi litre) 2 ounces = 1 pint

10 cl = 1 dl (deciliter) 2 pints = 1 quart

10 dl = 1 litre 4 quarts = 1 gallon

10 l = 1 dl (decaliter) 10 dl = 1hl(hectoliter)

Note :

10 hl = 1 kl (kiloliter) 1 gallon = 277.4 cubic inches

MENSURATION


2 VII - CLASS

§§ IMPORTANT :

Area is written in square units and volume in cubic units. To obtain them square or

cube. The measures of length in each length in each system.

Ex : 1 cm = 10 mm 1 sq. cm = 10 x 10 sq. mm 1 cubic cm = 10 x 10 x 10 cub. mm

1 foot = 12 inches 1 sq. ft. = 12 x 12 sq. in

1 are = 100 sq. m

1 hectare = 10000 sq. m

1 litre = 1000 cc.

A few equivalents : 1 inch = 2.54 cms. 1 m = 39. 37 inches

8 km = 5 miles 1 kg. = 2.2 pounds

Weight = Volume x Density

Imperal Units Conversion Facts

1 foot = 12 inches 1 kg is about 2.2 lbs.

1 yard = 3 feet 1 gallon is about 4.5 litres

1 pound (lb) = 16 ounces 1 litre is about 1.75 points

1 stone = 14 pounds 5 miles is about 8 km

1 gallon = 8 pints 1 inch is about 2.5 cm

1 foot is about 30 cm

§§ Plane figures :

i)Triangle :

Area = 1

2 base x height

For an equilateral triangle,

area = 23

4a

height 3

2h a

For a right angled triangle 1

2a b

a, b are legs and c = hypotenuse2 2 2c a b

If the angles in the right triangle are 45°, 45°, 90°, then the sides are in the ratio

1:1: 2 .

If the angles in the right triangle are 30°, 60°, 90°, then the sides are in the ratio

1: 3 : 2 .

ii) Quadrilateral :

Area = 1

2 one diagonal x sum of the perpendiculars from opposite vertices to the

diagonal.

= 1 2

1( )

2AC h h


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iii)For a parallelogram :

= base x heightiv) For a Kite & Rhombus :

1 2

1

2d d

v) For a rectangle: l b

Perimeter p = 2( )l b

vi)For a square : = (side)2

Perimeter p = 4a

vii) For trapezium :

1( )

2h a b

= 1

2 height x (sum of the parallel sides)

viii) Circle : 2A r

2p r

Area of a ring = 2 2( )R r

ix) Area of the track around a rectangle : Let l be the length and b be the breadth of a

rectangular field.

(a) A track of width d is formed around it out side the.

The outer dimensions are 2 , 2l d b d

Area of the track = 2( 2 )( 2 ) 2 ( ) 4l d b d lb d l b d

(b) A track of width d is found around it and inside :

The inner dimensions are 2 , 2l d b d

Area of the track = ( 2 )( 2 )lb l d b d

= 22 ( ) 4d l b d

x) Solids : Bodies that have a definite shape and occupy space are called solids.

We see objects like tables, chairs, books, pencils, pencil boxes, gas cylinders, balls, etc.

xi) Cuboid : A solid bounded by six rectangular faces is called a cuboid.

Ex : A box, a match box, a book, a brick, a tile, etc.

The above figure shows a cuboid. We name it as cuboid (ABCD, EFGH)

a) Faces of a cuboid : A cuboid has 6 rectangular faces are ABCD, EFGH,

A B

C D

E F

G H b

h

l


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EFBA,HGCD, FGCB and EHDA.

We can identify three pairs of opposite faces. They are

(i) The top face ABCD and the bottom face EFGH.

(ii) The front face EFBA and the back face HGCD.

(iii) The side faces FGCB and EHDA.

Note : Any two faces of the cuboid which are not the opposite faces are called adjacent

faces of the cuboid. For example EFBA and FGCB are two adjacent faces of the cuboid.

b) Edges of a cuboid : Two adjacent faces of a cuboid meet in a line segment

called an edge of the cuboid. A cuboid has 12 edges.

The edges of the given cuboid are AB, DC, HG, EF, AD, BC, EH, FG, AE, DH, BF and CG.

c) Vertices of a cuboid : The point of intersection of any three edges of a

cuboid is called a vertex. A cuboid has 8 vertices.

In the above figure, the vertices of the cuboid are A, B, C, D, E, F, G, H.

d) Base : The bottom face of a cuboid is called its base. In the given figure

EFGH is the base.

e) Lateral faces of a cuboid : The four faces which meet the base are called the

lateral faces of the cuboid. In the given figure EFBA, HGCD, FGCB and EHDA are lateral faces.

f) Dimensions of a cuboid : Since the opposite sides of a rectangle are equal,

we have

AB = DC = HG = EF = l, called the length of the cuboid

BC = AD = EH = FG = b, called the breadth of the cuboid

AE = DH = BF = CG = h, called the height (or) depth of the cuboid

g) Solid cuboid or Cuboidal region : The space bounded by the six faces of a

cuboid is called the cuboidal region .

Ex : Water in a rectangular tank, air in a room, a tile, a brick etc.

xii) Cube : A cuboid whose length, breadth and height are equal is called a cube.

Each edge of a cube is called its side.

Ex : Dice, ice cubes etc.

¶¶ Volume : The space occupied by a solid body is called its volume.

¶¶ Standard unit of volume : The standard unit of volume is 1 cubic centimetre,

written as 1 cu. cm or 1 cm3.

The volume of a cube of side 1 cm is 1 cm3.

¶¶ Volume of a cuboid : Consider a cuboid whose length = 4 cm,

breadth = 3 cm and height = 2 cm

Divide it into small cubes, each of side 1 cm, as shown in the following figure

A

D

H G

E

B

F

C


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Clearly, it is divided into 24 equal cubes, each of volume 1 cm3

The volume of the cuboid is 24 cm3

Also, (length × breadth × height)= (4 × 3 × 2) cm3

= 24 cm3

In general, we have, volume of a cuboid = (length × breadth × height) = l×b×h

¶¶ Conversion of Units of Volume :

(i) 1 cm3 = 1 cm × 1 cm × 1 cm

= 10 mm × 10 mm × 10 mm

= 1000 mm3

(ii) 1 dm3 = 1 dm × 1 dm × 1 dm

= 10 cm × 10 cm × 10 cm

= 1000 cm3 = l l

(iii) 1 m3 = 1 m × 1 m × 1 m

= 100 cm × 100 cm × 100 cm

= 1000000 cm3

= 1000 l = 1 kl

Note : 3 3 11000 1 1 1

1000cm l cm l

millilitre

1 l = 1 dm3 = 1000 cm3 = 1000 ml.¶¶ Surface area of a cuboid : The sum of the areas of all the 6 faces of a cuboid

is called its total surface area or simply its surface area.

¶¶ Lateral surface area of cuboid : The sum of the areas of all the four lateral

faces of a cuboid is called its lateral surface area.

Clearly, the lateral surface area is the sum of the areas of all faces excluding those

of the top face and the bottom face.

§§ To find the surface area and lateral surface area of a cuboid : Let us consider

a cuboid of length = l units. breadth = b units and height = h units

(i) Surface area of the cuboid

= sum of the areas of its 6 faces

= area ABCD + area EFGH + area FGCB + area EHDA

+ area EFBA +area HGCD

= lb lb bh bh lh lh sq. units

= 2 lb bh lh sq. units.

(ii) Lateral surface area of the cuboid

= area FGCB + area EHDA + area EFBA + area HGCD

l

b

h


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= bh bh lh lh sq. units

= 2 l b h sq. units.

(iii) For a cube of side a units, we have l = b = h = a.

Surface area of a cube = 6a2 sq. unitsLateral surface area of a cube = 4a2 sq. units

ACTIVITY - I

Find as many ways as you can

Each of the six triangles in the hexagon has the same dimensions.Calculate the

total area of the hexagon.

10cm

4cm

8cm

Method : 1

I can see six small triangles.

The base of each triangle is 5 cm, the height is 4 cm.

The area of a triangle is 1

base height2 .

Each of these triangles has an area of 215 4 10

2cm .

The area of the hexagon is 26 10 60cm .

Method : II

I can see a rectangle with four right angled triangles that have been removed.

The area of the rectangle is 10 x 8 = 80 cm2.

The base of each triangle is 2.5 cm. (10 5) 2 and the height is 4 cm.

The area of a triangle is 1

base height2 .

Each of the triangles has an area of 212.5 4 5

2cm .

The area of the hexagon is 80 – 20 = 60 cm2.Method : III

I can see two trapezia.


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The bases of the trapezia are 5 cm and 10 cm and the height is 4 cm.

The area of a trapezium is 1

( ) height2

a b .

Each of these trapezia has an area of 21(5 10) 4 30

2cm .

The area of the hexagon is 2 x 30 = 60 cm2.Method : IVI can see 12 small right - angled triangles

The base of each triangle is 2.5 cm, the height is 4 cm.

The area of triangle is 1 base height2 .

Each of these triangles has an area of 212.5 4 5

2cm .

The area of the hexagon is 12 x 5 = 60 cm2.

ACTIVITY - II

Tangram

Cut out the square below into 7 shapes.

This is a very old Chinese puzzle known as a tangram. Cut out the 7 shapes and

rearrange them to form :

a) a square from two triangles, and then change it to a parallelogram.

b) a rectangle using three pieces, and then change it into a parallelogram.

c) a trapezium with three pieces;

d) a parallelogram with four pieces;

e) a trapezium from the square, parallelogram and the two small triangles;

f) a triangle with three pieces;

g) a rectangle with all seven pieces.

Finally, put pieces back together to form the original square.


8 VII - CLASS

Example1: In rectangular farm ABCE, a tractor has ploughed plot BCD, the dimensions ofwhich are given in the adjacent figure. Find the area that remains to be ploughed.

260 m

10 m 240 m

A B

C D E

Sol : In right - angled BCD ,

Hypotenuse BD = 260 m and base DC = 240 m

2 2 2BC BD DC

= 2602 – 2402

= 67600 m2 – 57600 m2

= 10000 m2

BC = 100 m

Thus, area ploughed = 1

2 base × altitude

= 21

240 100 120002

m

= 1.2 hectares

Now length of the farm = 240 + 10 = 250 m

Breadth of the farm = 100 m

Area of farm = 250 m × 100 m = 2500 m

= 2.5 hectares.

Thus, area to be ploughed = 2.5 – 1.2

= 1.3 hectares

Example 2: Find the volume of the following solid shape. The solid is made up of

three cuboids 1, 2 and 3.

15 c

m 8 cm

1 2 3

8 cm

8 cm

EXAMPLES


9 VII - CLASS

Sol : We know that the volume of the cuboid = length × breadth × height

Volume (1) = 15 × 4 × 24 (h = 8 + 8 + 8)

= 1440 cm3

Volume (2) = 15 × 4 × 16 (h = 8 + 8)

= 960 cm3

Volume (3) = 15 × 4 × 8 (h = 8)

= 480 cm3

Total volume = 1440 + 960 + 480

= 2880 cm3

Example 3: The rain fall on a certain day was 4 cm. How many litres of water fell on 2hectares of a field on that day ?

Sol : The area of the field = 2 hectares

= 22 10000 m

= 22 10000 100 d m

The depth of rain water on the field = 4 cm

= 4 2

10 5dm dm

The volume of rain water on the field

= (area of the field) × (depth of water on it)

= 32

2 10000 1003

dm

= 800000 dm3

= 800000 litres

= 58 10 litres

The volume of water that fell on the field on that day was 58 10 litres.

Example 4 : Estimate area of each of the following shapes. The side of the square is 1 unit.

Sol : Diameter of the circle = length of the side of the square

12 1

2r r

(i) Area of the shaded portion= Area of the square – 2 x area of the semi circle

= 2

2 1 1(1) 2

2 2


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= 21 1

1 2 12 4 4

m

.

(ii) Area of the shaded portion= Area of the square-Area of the circle

= 2

2 1(1)

2

= 21

4m

.

Example 5 : Find the area of each shape below.

Sol : (i) The given area

= Area of the rectangle + Area of the triangle

Area of rectangle = l x b = 210 8 80cm

The base of the triangle b = 10 cm, height = h = 12–8 = 4 cm

Area of triangle = 21 110 4 20

2 2b h cm

Area of the given figure = 280 20 100cm

(ii) The given area can be divided into 4 triangles

Total area = 1 1

2 2x z y z

1 1

2 2x t y t

= 1 1

( ) ( )2 2

z x y t x y

= 1 1 1

( )( ) 3 4.82 2 2

x y z t BD AC

= 23 2.4 7.2cm

Example 6 : A wheel makes 10,000 revolutions in covering a distance of 8.8 km. Find theradius of the wheel.

Sol: Let r = radius of the wheel.

Distance covered in one revolution 22

2 27

r r

Distance covered in 10,000 revolutions = 44

100007

r


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Given that 44

10000 8.87

r km = 8800 m = 880000 cm

4488

7r

88 714

44r cm

.

Example 7 : Three equal cubes of side 6 cm are joined end to end. Find (i) the volume (ii)the surface area, and (iii) the lateral surface area of the resulting cuboid.

Sol : Clearly the resulting cuboid has length = 18 cm,

breadth = 6 cm, height = 6 cm

(i) volume of the cuboid = l b h cubic units

= (18 × 6 × 6) cm3

= 648 cm3

(ii) Total surface area of the cuboid = 2 lb bh hl square unitss

= 2 18 6 6 6 6 6 square. cm

= 2 (180) cm2

= 360 cm2

(iii) The lateral surface area of the cuboid = 2 l b h sq. units

= 22 18 6 6 cm

= 288 cm2

Example 8 : If the length and breadth of a rectangular room are each increased by 1 m,then the area of the floor is increased by 21 sq. m. If the length is increased by 1 m andbreadth is decreased by 1 m, then the area is decreased by 5 sq. m. Find the area ofthe room.

Sol : Let l = length

b = breadth of the room

Area = A l b

When the length, breadth are increased by 1 m, the area is increased by 21 sq.m

( 1)( 1) 21l b A

1 21 20lb l b lb l b ------------(1)

When the length is increased by 1 m and breadth is decreased by 1 m, the area isdecreased by 5 sq. m.

( 1)( 1) 5l b A

( 1)( 1) 5l b lb

1 5lb l b lb

4l b ----------------(2)

Now (1) + (2) 20 4l b l b

2 24

12

l

l


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Now (1) – (2) 20 4l b l b

2 16b

8b

12 8 96 .Area A l b sq m

Example 9: A cow is tethered in the middle of a circular field with 14 feet long rope. If thecow grazes 90 sq feet each day, then find approximately the number of days needed tothe cow to graze the whole field.

Sol : Radius r = 14 feet

Area = 2 2214 14 44 14 616 .

7A r sq feet

Number of days needed by the cow to graze the field completely =

6167

90 . 90

Area

sq feet days

Example 10 : A metal cube of edge 24 cm is melted and formed into three smaller cubes. Ifthe edge of two small cubes are 12 cm and 16 cm, find the edge of the third small cube.

Sol : The volume of the small cube which has edge 12 cm is 3 312 1728cm

The volume of the small cube which has edge 16 cm is 3 316 4096 cm

The sum of the volumes of the two smaller Cubes = 1728 + 4096 = 5824 cm3The volume of the metal cube = 243

= 13824 cm3

The volume of the third smaller cube = 13824 – 5824

= 8000 cm3

The edge of the third cube = 20 cm (8000 = 20 × 20 × 20)

I) MCQ’s wih Single correct answer type :

1. If the base of triangle 24 cm and the corresponding height 14.5 cm then area of triangle

A) 168cm2 B) 174cm2 C) 186cm2 D) 182cm2

2. If area of equilateral triangle is 36 3 cm2 then its perimeter is ...cm

A) 44 B) 42 C) 45 D) 48

3. If area of equilateral triangle is 36 3 cm2 then its height is ...cm

A) 6 3 B) 3 3 C) 4 3 D) 9 / 3

4. The length of the three sides of a triangle are 20cm, 16cm, 12cm then its area is ....cm2

A) 96 B) 120 C) 144 D) 160

5 The base of a triangle is 5 cm and its area is 20cm2 then the height of the triangle is .......

A)8cm B)9cm C)10cm D)12cm

6. The circumference of a circle of radius 10.5cm is .......

A) 66cm B) 68cm C) 346.5cm2 D) 364.5cm2

7. The difference between circumfrence and radius of a circle is 37 cm, then area of circle

TEACHING TASK


13 VII - CLASS

is ___

A) 111 cm2 B) 148cm2 C) 154 cm2 D) 259 cm2

8. In a triangle one side is 16 cm and its corresponding height is 9 cm, then its area is ___

A) 72 cm2 B) 14.4 cm2 C) 48 cm2 D) 28 cm2

9. The area of square of side 20 cm is same as the area of rectangle length 40 cm, thenbreadth of rectangle is____

A) 20cm B) 10 cm C) 15 cm D) 25 cm

10. The area of a rhombus having each side 13cm and one of its diagonals is 24cm is..cm2

A) 110 B) 100 C) 108 D) 120

11. The lengths of the diagonals of the rhombus are 16cm and 12cm .The length of eacheach side of the rhombus is .


12. There is a rectangular field opf length 560 m and breadth 35 m. There is a another fieldin the shape of square having the same area as the rectangular field. Find the perimeterof square field.

A) 510 m B) 520 m C) 540 m D) 560 m

13. The length and breadth are in the ratio 5:3, if its perimeter is 192 cm, find its area.

A) 2160 cm B) 2016 cm C) 2017 cm D) 2170 cm

II) MCQ’s with more than one correct answers :

This section contains multiple choice questions. Each question has 4 choices (A), (B), (C),(D),

out of which ONE or MORE is correct. Choose the correct options

14. The sides of the triangle are in the ratio 5:12:13 and its perimeter is 150m then its sidesare

A) 25cm B) 60cm C) 65cm D) 38cm

15. The ratio sides of a triangle is 1:1: 2 , then the ratio of the angles opposite to the those

sides is___

A) 1:1:2 B) 1:2:3 C) 2:2:4 D) 8:8:16

16. If the angles of a tringle are in the ratio 1:1:1, then ratio of the sides of the tringle is___

A) 1:2:3 B) 1:1:1 C) 100:100:100 D) 1:1: 217. In square the ratio of side and its diogonal is ____

A) 3 :1 B) 3 : 6 C) 2 :1 D) 1: 218. The base af a parallelogram is twice its height, if the area is 98cm2 then its base and

height respectively are

A) 7cm B) 8cm C) 14cm D) 16cm

19. The difference between the perimeter of a rectangle 15 cm by 10 cm and side of square

is 12 cm?

A) 2 cm B)12 cm C) 1 X2 cm D) 25 cm

III) Assertion and Reasoning type questions:

This section contains certain number of questions. Each question contains Statement – 1 (Assertion) andStatement – 2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE iscorrect Choose the correct option.

a) Both A and R are correct and R is correct explanation of A.

b) Both A and R are correct and R is not correct explanation of A.


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c) A is correct and R is incorrect. d) A is incorrect and R is correct.

20. A: The ratio of a circumfrence of a circle and its diameter is equal to 22:7

R: Circumference of circle: 2 r

21. A :The diagonal of square is ‘s’ units, then its area is 2

2

s units

R: Area of Square = 2( )

2

diagonal sq.unitss

IV) Comprehension :

This section contains paragraph. Based upon each paragraph multiple choice questions have to beanswered. Each question has 4 choices (A) , (B) ,(C ) and (D) out of which ONLY ONE is correct. Choosethe correct option.

The length of boundary of a closed figure is called a perimeter. The amount of surfaceenclosed by a closed figure is called its area

22. The piece of a wire is 100 cm. If it is converted square shape, then are of its square shape

is___

A) 625 cm2 B) 630 cm2 C) 570 cm2 D) 324 cm2

23. The perimeter of rectangle is ‘p’ units and breadth is half of its length, area of rectangle

is___

A) 2

6

pB)

2

18

pC)

2

12

pD)

2

30

p

24. How many tiles with dimensions 12 cm, 5 cm. We need to fit a region whose length and

breadth are 144 cm and 100 cm.

A) 220 B) 240 C) 140 D) 340

V) MATCH THE FOLLOWING:

This section contains Matrix-Match Type questions. Each question contains statements given in twocolumns which have to be matched. Statements (A, B, C, D) in Column–I have to be matched withstatements (p, q, r, s) in Column–II. The answers to these questions have to be appropriately bubbledas illustrated in the following example.If the correct matches are A-p,A-s,B-r,B-r,C-p,C-q and D-s,then the correct bubbled 4*4 matrix shouldbe as follows:

25. COLUMN-1 COLUMN-II

i) The length of three sides of triangle are 26cm, 28cm,30cm, the corrresponding height to the base 28cm is .....cm a) 6ii) The area of an equilateral triangle is

4 3 cm2 then its perimeter is ...cm b) 4

iii) If the height of equilateral triangle is 3. 3 cm

then its each side is ....cm c) 24iv) The base of an isosceles tringle is 6cm and each of equal sides is 5cm then its height is ....cm d) 12

A) i-a, ii-b, iii-c, iv-d B) i-a, ii-b, iii-c, iv-d

C) i-c, ii-d, iii-a, iv-b D) i-b, ii-d, iii-a, iv-c


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26. Cloumn - I Cloumn - II

i) The perimeter of one face of square stone is

36 cm then length of side is a) 12 cm

ii) The area of square field 72 cm2 , then diaognal is b) 9 cm

iii) No. of different rectangles with integral measurments

can be drawn with the perimeter as 32 m. c) 1

iv) One square decameter equal to ___ are d) 6

e) 3

A) i-a, ii-b, iii-c, iv-e B) i-c, ii-b, iii-e, iv-a

C) i-b, ii-a, iii-d, iv-c D) i-a, ii-a, iii-d, iv-e

VI) Solve the following :

1. The lengths of the sides of the triangle are three consecutive odd numbers. The shortestside is 20% of the perimeter. What is the perimeter of the triangle?

2. The base of triangle is 15 cm and height is 12 cm, then find the height of another triangledouble the area having the base 20 cm.

3. Find the area of parallelogram ABCD in which AB=60cm, BC=40cm, AC=80cm

( 15 3.87 )

4. The base and height of a triangle are in the ratio 4:5.If its area is 250m2,find its base andthe height.

5. A rectangular piece of plastic sheet measures 65cm by 40cm. Find its cost at the rate ofRs 20 per sq meter.

6. If the perimeterof isosceles right angle triangle is 2014cm then find its area.

7. A recangular plot measuring 120m by 60m is surrounded by a path 5m wide.Find thecost of paving the path with gravel at the rate of Rs75 per sq.m.

I) MCQ’s with single correct answers :

1. The base of a triangle is 3dm and its area is 18cm2 then its altitude is ....cm

A) 1.2 B) 0.5 C) 0.6 D) 6

2. Base of a triangle is 25cm and corresponding height is 10.8cm then its area is .......cm2

A) 135 B) 145 C) 154 D) 130

3. The sides of lengths of a triangle are 52cm, 56cm, 60cm repectively then its area is...cm2

A) 1340 B) 1344 C) 1433 D) 1430

4. The sides of lengths of a triangle are 52cm, 56cm, 60cm resly then its

perimeter is

A) 188cm B) 160cm C) 168cm D) 186cm

5. The height of an equilateral triangle is 6cm then its area is ....cm2

A) 20.87 B) 20.78 C) 27.08 D) 28.08

6. In isosceles triangle each of whose equal side is 13cm and whose base is 24cm then itarea is

A) 56cm2 B) 40cm2 C) 60cm2 D) 64cm2

BEGINNERS ( Level - I )

LEARNER’S TASK


16 VII - CLASS

7. The base of an isosceles triangle is 8cm long and each of its equal sides measures

6cm then its area is ......cm2

A) 16 5 B) 8 5 C) 16 3 D) 8 3

8. The area of a parellelogram is 50 cm2.If the base is 10 cm then find its correspondingheight.

A)10cm B)15cm C)5cm D) 25cm

9. The area of a parelellogram is 144cm2 and its height is 18cm, then the length of thecorresponding base.


10. Find the area of a parellelogram whose base is 3.5m and height 80 cm is.......A)2.8m2 B)8.3m2 C)3.5m2 D)8.5cm2

11. The area of a rhombus if the lengths of whose diagonals are 25cm and 16.8cm is ....cm2

A) 120 B)160 C) 190 D) 210

12. Which of the following is not true for a parellelogram?

A)opposite sides are equal. B) opposite angles are equal.

C) opposite angles are bisected by the diagonals

D) Diagonals bisect each other.

13. The area of a rhombus is 24cm2 .If one diagonal is 8cm, then its perimeter isA) 6cm2 B) 8cm2 C) 5cm2 D) 20cm2

Solve the following :

1. Find the altitude and area of an equilateral triangle of side 10cm.

2. How many different isosceles triangles of perimeter 20 cm exist with sides of integrallength?

3. The circumfrence of a circle 5.28 cm more than the radius find the diameter.

4. Find area of square inscibed in a circle of diameter 10 cm.

5. Find the area of trepezium whose parallel sides lengths are 5m, 3m,and its height is 4m.

6. Calculate the area of trepezium whose parallel sides lengths are 40cm and 21cm,and itsdiagonal is 29cm.(trapezium ABCD right angle at C).

I) MCQ’s with more than one correct answers :

This section contains multiple choice questions. Each question has 4 choices (A), (B), (C),(D), out of

which ONE or MORE is correct. Choose the correct options

1. The area of triangle whose sides are 6 cm, 8 cm, 10 cm-s___

A) 22 144 cm B) 2576 cm C) 224 cm D) 2124 cm

2. The perimeter of an Equilateral triangle is ‘a’ units then the length of each is _

A) 3a B) 3+a C) Quotient of ‘a’ by 3 D) a/3

3. The diagonal of a sqare is 9 2 cm .then its perimeter

ACHIEVERS ( Level - II )

EXPLORERS ( Level - III )


17 VII - CLASS

A)12 X 3cm B)36cm C)38cm D) 4 X 9cm

4. The perimeter of a rhombus is 52cm, if one diagonal is 10cm ,then the other diagonal is:

A)5.2cm B) 2 X12 cm C)22cm D)24cm

II) Assertion and Reasoning type questions :

This section contains certain number of questions. Each question contains Statement – 1 (Assertion) andStatement – 2 (Reason). Each question has 4 choices (A), (B), (C) and (D) out of which ONLY ONE iscorrect Choose the correct option.

a) Both A and R are correct and R is correct explanation of A.

b) Both A and R are correct and R is not correct explanation of A.

c) A is correct and R is incorrect. d) A is incorrect and R is correct.

5. Assertion :If side of equilateral triangle 4 cm, then its are is 3 cm2

Reasoning::Area of equilateral triangle=3

42a sq. units

6. Assertion: The perimeter of regular hexagon of each side is ‘p’ units is 6 X p

Reasoning: The perimeter of polygon is sum of all sides.

III) Comprehension :

This section contains paragraph. Based upon each paragraph multiple choice questions haveto be answered. Each question has 4 choices (A) , (B) ,(C ) and (D) out of which ONLY ONE is correct.Choose the correct option.The side of square is 4 cm

7. The diagonal of square is_____

A) 4 2 cm B) 8 2 cm C) 9 2 cm D) 5 2 cm

8. The perimeter of square is___

A) 10 cm B) 12 cm C) 16 cm D) 14 cm

9. Area of square is ____

A) 16 cm2 B) 20cm2 C) 19 cm2 D) 25 cm2

IV) MATCH THE FOLLOWING:

This section contains Matrix-Match Type questions. Each question contains statements given in twocolumns which have to be matched. Statements (A, B, C, D) in Column–I have to be matched withstatements (p, q, r, s) in Column–II. The answers to these questions have to be appropriately bubbledas illustrated in the following example.If the correct matches are A-p,A-s,B-r,B-r,C-p,C-q and D-s,then the correct bubbled 4*4 matrixshould be as follows:

10. COLUMN-1 COLUMN-II

i) Area of an Equilateral triangle is a) 77 m2

ii) Area of right angled triangle b) s s a s b s c

iii) Heron’s formula for area of triangle c) 23

4side

iv) The area of semicircle of d=14 m d)1

2base height

e) 144 m2

A) i-a, ii-b, iii-c, iv-d B) i-a, ii-e, iii-c, iv-a


18 VII - CLASS

C) i-d, ii-b, iii-a, iv-e D) i-c, ii-d, iii-b, iv-a

11. Cloumn - I Cloumn - II

a)Area of square i)b h

b)Area of quadrilateral ii) 1

2h a b

c)Area of rhombus iii) 1 2

1

2d h h

d)Area of parallelogram iv) 1 2

1

2d d

v) a2

A) a-i, b-ii, c-iii, d-v B) a-v, b-iii, c-iv, d-i

B) a-v, b-iv, c-iii, d-i C) a-v, b-i, c-iv, d-iii

TEACHING TASK :1.B 2.D 3.A 4.A 5.A 6.A 7.D 8.A 9.B 10.D

11.A 12.D 13.A 14.A,B,C 15.A,C,D 16.B,C 17.B,D

18.A,C 19.A,C 20.B 21.A 22.A 23.B 24.B25.C 26.C

VI) 1.15cm 2.18cm 3.2322cm2 4.20m , 25 m

5.Rs.4200 6. 23 2 1007 cm2 7.Rs.142500

LEARNER’S TASK :BEGINNERS :

1.A 2.A 3.B 4.C 5.B 6.C 7.B 8.C 9.C 10.A

11.D 12.D 13.C

ACHIEVERS : 1)5 3,50 3 2)6 3)8.4cm, 16.8cm

4)50cm2 5)16m2 6)610cm2

EXPLORERS : 1.A,B,C 2.C,D 3.B,C,D 4.B,D 5.D 6.B

7.A 8.C 9.A 10.D 11.B

§§ MENSURATION - 3D

1. Lateral surface area of cuboid = 2h(l+b) sq. units

2. Total surface area of cuboid = 2(lb+bh+lh) sq. units

3. Volume of a cuboid = lbh cu. units

4. Length od diagonal of a cuboid = 2 2 2l b h units

5. Lateral surface area of cube = 4 2a sq. units

6. Total surface area of cube = 6 2a sq. unitss

7. Volume of a cube = 3a cu. unitss

KEY


19 VII - CLASS

8. Curved surface area of Cylinder = 2 rh sq. units

9. Total surface area of Cylinder = 2 r(r h) sq. unitss

10. Volume of Cylinder = 2r h cu. unitss

11. Curved surface area of Cone = rl sq. units

12. Total surface area of Cone = r(r l) sq. unitss

13. Volume of cone = 21r h

3 cu. unitss

14. Surface area of Sphere = 4 2r sq. unitss

15. Curved Surface area of HemiSphere = 2 2r sq. unitss

16. Total Surface area of HemiSphere = 3 2r sq. unitss

17. Volume of Sphere = 34r

3 cu. unitss

18. Volume of Hemisphere = 32r

3 cu. unitss

§§ PRISM :

A solid having two congruent and parallel faces clled bases and whose other faces, the

lateral faces are parallelograms formed by joining the corresponding vertices of the bases is

called a Prism.

§§ RIGHT PRISM :

A prism whose bases are perpendicular to the lateral edges and all lateral faces are

rectangles is called Right Prism.

A prism is named according to the shape of the base,

(I) If the base of a prism is tiangle then the prism is called triangular prism

(II) If the base of a prism is rectangle then the prism is called rectangular prism (or)Cuboid.

(III) If the base of a prism is Square then the prism is called square prism or Cube.

a) Area : The measure of the surface enclosed by a plane figure is called its “Area”. Area is

measured in “square”units.

b) Volume: The space occupied by an object is called its volume.Volume is measured in

cubic units.

c) Area of Four walls of room:

Let there be a room with Length = l units, breadth = b units and height = h units.Then,

i) Area of 4 - walls = xhbl2 sq.unitss

ii) Diagonal of the room = 222 hbl units.

§§ Solids: The bodies occupying space are called solids. The solid bodies occur in

various shapes. Such as a “cuboid” a cube, a cylinder, a cone and a sphere etc.

¶¶ Volume of a solid: The space occupied by a solid body is called its volume. The

units of volume are “cubic cm” (ic cm3) (or) cubic metres (ic m3) etc.

§§ CUBOID

A right prism whose base is a rectangle is called Cuboid (or) A cubiod is a solid bounded

by six rectangular plane regions.


20 VII - CLASS

Some examples of cuboid are Match box, Duster, Book etc.

In general a cuboid has its length, breadth and height of different values (sizes).

The above figure shows a cuboid. It is clear that a cuboid will have

a) Six faces :

Namely ABCD, ACEH, CDFE, ABGH, BDFG, EFGH,

b) Twelve edges :

AB, AC,CD, BD, Ec, df, Ef, EH, FG, GH, BG, AH

c) Eight corners :

A,B,C,D,E,F,G and H

If a cuboid having length, breadth and height are l,b, h respectively, Then

Lateral surface area (LSA) = Perimeter of basse x height

= 2 (l+b)x h

= 2h (l +b) sq units

Total surface area (TSA) = LSA + Area of Top +Area of botten

= 2h (l+b)+lb+lb

= 2 lh +2bh +2lb

= 2( ld + bh +lh) sq units.

Volume = Area of base x height

= lb x h

= lbh cu. units

Note : 1) Length of diagonal = 222 hbl units

2) Length of edges = 4 ( l+b+h) units

§§ CUBE :

A cuboid in which length, breadth and height are equal is called a cube.

An examples for a cub is dice.

Each side of a cube is called its edge. Thus all edges of a cube are equal.

Clearly, a cube has 6 faces, 12 edges and 8 vertices.f the edge of a cube is ‘a’ units.Then

Lateral surface area = perimeter of base x height

= 2 (a+a)xa

= 2(2a) x a

= 4a2 sq units

Total surface area = LSA + Area of Top + Area of botton

= 4a2 +a2 + a2

= 6a2 sq units

Volume = Area of base x height

= (ax a) xa

=a3 cu. units.


21 VII - CLASS

Note : 1) Length of diagonal = 3 a units

2) Sum of edges = 12a units

I) MCQ’s with single correct answer type :

1. The base of a right prism is an equilateral triangle. If the volume and lateral surface areaof prism be 60 3 cm3 and 180 cm3 respectively. Then the lengh of each side of its baseis -------cm

a) 1 b) 2 c) 3 d) 4

2. If the dimensions of a cuboid are in the ratio of 1:2:3 and its T.S.A is 88 cm2, then thevolume of a cuboid is __________ cm3

a) 84 b) 32 c) 72 d) 48

3. The base of a right prism is a right angled triangle. The perimeter of this triangle

measures 60 cm and its hypotenuse measures 25cm. If the height of the prism is

18 cm. Then its volume is _________________ cm3.

a) 2700 b) 2500 c) 2400 d) 3200

4. The area of the floor of a room is 15cm2 If its height is 4m. Then the volume of the aircontained in the room is _________lm3

a) 60 b) 600 c) 6000 d) 60000

5. If the volume of a cuboid is 440 cm3 and the area of its base is 88cm2. Then its height is__________cm

a) 5 b) 9 c) 11 d) 4

II) MCQ’s with More than one answer type:



1. If three cubes of sides 3cm, 4cm, and 5cm are melted and a new cube is formed. Thenthe side of the new cube is __________cms

a)3x2 b) 4/2 c) 5x2 d) 6

2. Three equal cubes are placed adjacently in a row. Then the ratio of total surface area ofthe new cuboid to that of the sum of the surface areas of the three cubes is

a) 7:8 b) 7:9 c)9:7 d) 14:18

3. The length of the diagonal of cuboid whose dimensions are 3cm x 4cm x 5 cm is..

a) 50 cm b) 5 2 cm c) 2 5 cm d) 4 5 cm

III) Matrix Matching :


1. Column - I Column - II

i) 1 dm2 P) 1000000 m2

ii) 1 km2 Q) 10000 cm2

TEACHING TASK


22 VII - CLASS

iii) 1 cm2 R) 100 cm2

iv) 1m2 S) 100 mm2

a) i) -S, ii) R, iii) P, iv) Q b) i) R, ii) P, iii) S, iv) Q

c) i) S, ii) Q, iii) P iv)R d) i) P, ii) R, iii) S iv) Q

IV) Comprehension type :

This section contains paragraph. Based upon each paragraph multiple choice questions have to beanswered. Each question has 4 choices (A) , (B) ,(C ) and (D) out of which ONLY ONE is correct. Choosethe correct option.If the side of a cube is 5 cm. Then

1. The length of the diagonal is ___________ cm

a) 153 b) 152 c) 53 d)15

2. The lateral surface area is ______cm2

a) 75 b) 100 c) 125 d) 150

3. The total surface area is _______cm2

a) 75 b) 100 c) 125 d) 150

4. The volume of the cube is __________cm3

a0 75 b)100 c) 125 d) 150

I) MCQ’S with Single coorect answer type:

1. If the dimensions of a cuboid are 10 x 8 x 5cm. Then its lateral surface area is ____cm2

a) 180 b) 200 c) 280 d) 290

2. The surface area of a cuboid is 100cm2 and the lenght and height are 7cm, 2cm. Thenthe breadth of the cuboid is __________cm

a) 3 b) 4 c) 5 d) 6

3. If the volume of a cube is 216cm3. Then its surface area is __________cm2

a) 36 b) 64 c) 216 d) 256


1. An olympic suming pool is in the shape of a cuboid of dimensions 50 m long 25 m wide.If it is 3m depth, throughout, how many liters of water does it hold?

2. A classroom is 7m long, 6.5 m wide and 4m high. It has one door 3m x 1.4 m and threewindows each measuring 2m x 1 m. The interior wall is to be colour washed. the

contractor charges Rs. 5.25 per sq.m Find the cost of colour washing of such 15 walls.

3. A cuboided tin is 30m by 40m by 50m. Find the cost of the tin required for making 20 suchtins if the cost of tin sheet is Rs. 20 per sq. mts.

4. A metal cube of edge 12cm is metted and formed into three smaller cubes. If the edges oftwo smaller cubes are 6cm and 8cm. Find the edge ofthe third smaller cube.

5. If the ratio of volumes of two cubes is 1:8. then find the ratio of resulting diaganals of thosecubes.

LEARNER’S TASK

BEGINNERS( Level - I )



23 VII - CLASS

I) MCQ’S with More than one answer type:



1. The surface area of a cube is half of its volume.Then the length of the edge is___cm

a) 8 b) 4x2 c) 12 d) 6x2

2. One edge of a cube is 4cm. Then its base perimeter is

a) 8 b) 16 c) 4*4 d) 32/2

II) Matrix Matching :

This section contains Matrix-Match Type questions. Each question contains statements given in twocolumns which have to be matched. Statements (A, B, C, D) in Column–I have to be matched withstatements (p, q, r, s) in Column–II. The answers to these questions have to be appropriately bubbledas illustrated in the following example.If the correct matches are A-p,A-s,B-r,B-r,C-p,C-q and D-s,then the correct bubbled 4*4 matrix shouldbe as follows:Column - I Column - II

i) No.of faces of a cuboid P) square

ii) No.of edges of a cuboid Q) rectangle

iii) No.of vertices of a cuboid R) 6

iv) The shape of face in a cuboid S) 8

T) 12

a) i) -T, ii) S, iii) R, iv) P b) i) S, ii) T, iii) R, iv) P

c) i) R, ii) T, iii) S iv) Q d) i) R, ii) T, iii) S iv) P

III) COMPREHENSIONType :

This section contains paragraph. Based upon each paragraph multiple choice questions have tobe answered. Each question has 4 choices (A) , (B) ,(C ) and (D) out of which ONLY ONE is correct.Choose the correct option.

A) If the length, breadth and height of a cuboid are 5 cm, 4 cm and 3 cm respectively.Then

1. The length of the diagenal is ___________cm

a) 60 b) cm3

210c) cm25 d) cm5

2. The total surface area of cuboid is ______cm2

a) 54 b) 154 c) 144 d) 94

3. The volume of a cuboid is __________cm3

a) 60 b) 180 c) 160 d) 240

MCQ’S with Single coorect answer type:

1. The height of a wall is six times it width and the length of the wall is seven times its height.If volume ofthe wall be 16128 c.c its width is ______m (C.B.I.1998)

a) 4 b) 4.5 c) 5 d) 6

2. The dimenstions of two cuboids are 10cm, 8cm, 6cm, 15cm, 12cm, 9cm respectively.Then the ratio of their volume is ________ (NTSE)

a) 8:27 b) 27:8 c) 16:25 d) 25:16


RESEARCHERS ( Level - IV )


24 VII - CLASS

TEACHING TASK :I. 1.D 2.D 3.A 4.A 5.A

II. 1.A,D 2.B,D 3.A,B

III. 1.B

IV. 1.C 2.B 3.D 4.B

LEARNER’S TASK : BEGINNERS :

LEVEL - 1

I. 1.A 2.B 3.C

EXPLORERS:

I. 1.C,D 2.B,C,D II. 1.C

III. 1.C 2.D 3.A

RESEARCHERS :

1.A 2.A

§§ CYLINDER :

If the line joining the centres of the circular bases is perpendicular to the base,then the

solid figure is called a right circular cylinger (or) A right prism whose base is a circle is called

cylinder.

The figure below shows a cylinder of radius ‘r’ and height ‘h’

When a right circular cylinder is cut vertivally and unfolded then it forms a rectangel.

Curved surface area = A rea of rectangle

= Lenght x breadth

= 2pr x h

= 2prh sq units.

Total surface area = curved surface area + area of top + area of botom

= 2prh +pr2 + pr2

= 2prh +2pr2

= 2p r(r+h) sq units

Volume of cylinder = Area of base x height

= pr2x h

= pr2 h cubic units

KEY


25 VII - CLASS

Note :

If the heights of two cylinders are equal, then thier curved surface areas are in the ratio

of their radii and volumes are in the ratio of squares of their radii.

If the base radii of two cylinders are equal, then thier curved surface areas and volumes

are in the ratio ofthier heights.

If the curved surface area of two cylinders are equal, then thier radii and height are in

inverse ratio.

§§ CONE :

A solid having lateral surface as curved surface and one end as circular plane surface

and other end as verten is called cone. (or)

A cone is generally defined as the solid generated by the revolution of a right angled

triangle about one of its sides, containing right angle.

An example of cone is ice cream cone.

The figure below repersents a cone of radius ‘r’ height ‘h’ and slat height ‘l’.

When a cone is unfold, it forms a sector. so, the curved surface area of cone is

equal to the area of a sector.

Curved surface area = sum of areas of triangles formed in a sector

= 21

xl (2pr)

= prl sq units

Total surface area = curved surface area + Area of basse

= prl + r2

= pr(r+r) sq units

Volume of cone = 3

1x volume of cylinder

= 3

1x p r2h cubic units

Note :

Slant height, l= 22 hr


26 VII - CLASS

I) MCQ’s with single correct answer :

1. Two cylindrical jars have their diameters in the ratio 3:1, but height 1:3, then the ratio oftheir volumes is _____________

a)1:4 b) 1:3 c) 3:1 d) 2:5

2. The ratio of the curved surface area to the total surface area of a right circular cylinder is1:2, the TSA is 616 cm2 then volume of the cylinder is ......... cm3

A) 1078 B) 1780 C) 1870 D) 1087

3. The diameter of a garden roller is 1.4 m and it is 2 m long. The area covered by the rollerin 5 revolutions is ......... m2.

A) 64 B) 34 C) 82 D) 44

4. 50 circular plates each of radius 7 cm and thickness 0.5 cm are placed one above theother to form a right circular cylinder. Then its total surface area is ...... cm2

A) 1408 B) 1804 C) 1480 D) 1084

5. If the diameter of the cross-section of a wire is decreased by 5 %. The percentage

increase in the length so that the colume remains the same is .......

A) 10.8 B) 12.4 C) 11.5 D) 18.3

6. The volume of the largest circular cone that can be out of a cube whose edge is 14cm is

A) 708.68 cm3 B) 718.66 cm3 C) 716.88 cm3 D)788.68 cm3

II) More than one answer type:



1. A sector of a circle of radius 12 cm has an angle 120o. It is rolled up so that twobounding radii are joined together to form a cone. Then the volume of the cone is ...cm3

A) 1823

B) 64 2 2

3

C) 128

3

D) 128 2

3

2. The radius of two cones are in the rario of 1:2 and their salnt heights are in the ratio 3:5.Then the ratio of their curved surface areas is ...........

A) 3:10 B) 10:3 C) 30:100 D) 4:9



1) Column - I column - II

i) If the volume of the cylinder is 68.75cc

and its radius is 2.5cm then its height is P) 66cm

ii) If the volume of a right circular cylinder

with its height equal to the radius is 1

257

3cm ,

then the radius of the cylinder is .... Q) 7cm

TEACHING TASK


27 VII - CLASS

iii) The total surface area of a right circular

cylinder is 1683 2cm and its height is 15cm

then the perimeter of the base is ..... R) 3.5cm

iv) If the volume of the cylinder is 88 cc and radius is 2cm then its height is ..... S) 2cm

T) 4cm

a) i) T, ii) S, iii) R, iv) P b) i) S, ii) T, iii) R, iv) P

c) i) R, ii) S, iii) P, iv) Q d) i) R, ii) T, iii) S iv) P

IV) Comprehension Type:


. A right circular cylinder has base radius 14cm, and height 21 Cm. Then

1. Area of base is ............ cm2

A) 616 B) 606 C) 1844 D) 1484

2. Curved surface area is ............ cm2

A) 1488 B) 1848 C) 1844 D) 1484

3. Total surface area is ............ cm2

A) 3808 B) 3800 C) 3080 D) 3008

4. Volume of cylinder is ............ cm3

A) 13969 B) 12936 C) 13966 D) 13696

I) MCQ’s with Single coorect answer type :

1. The total surface area of a cylinder whose base circumference is 154cm and height15 cm is ........... cm2

A) 2310 B) 2130 C) 6083 D) 6038

2. The vertical cross section of a right circular cylinder is

A) Rectangle B) Circle C) Triangle D) Parallelogram

3. If the volume of a cylinder is 308 cm3, its height is 8 cm, then the diameter of the base is........ cm

A) 5 cm B) 28 cm C) 7 cm D) 10cm


1. A rectangular sheet of paper 44 cm X 20 cm is rolled along its length and a cylinder isformed and again rolled along its breadth another cylinder is formed. Find the differenceof their volumes.

2. The ratio of two right cylinders ae in the ratio 2:3 and thier heioghts are in the ratio 5:3.Calculate the ratio of thier curved surface areas and the ratio of their volumes.

3. The difference between the outside and inside surface of a cylindrical metallic pipe 14cm is 44 sq.cm. If the pipe is made of 99 cubic cm of metal, find the outer and inner radiiof the pipe.

LEARNER’S TASK




28 VII - CLASS

4. The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete

revolutions to roll once over the play ground to level. Find the area of the play ground in m2

5. Two similar cones have volumes 12 cm. inits and 96 cm.units. If the curved

surface area of the smaller cone is 15 sq.units. What is the curved surface area of thelarger one ?

6. The curved surface area of a cone is 5

11597

cm2. Area of its base is 4

2577

cm2.Find itss

volume.

7. The circumference of the base of a 10 m height conical tent is 44 mts. Calculate thelength of canvas used in making a tent if width of canvas is 2 m.

8. The radius and height of a cone are in the ratio of 4:3 and area of base is 154cm2. Thenfind its slant height and curved surface area.

9. A cone and a cylinder have equal bases and heights. Then find the ratio of their

corresponding volumes.

I) MCQ’s with More than one answer type:



1. The base area of a cone is 38.5 cm2 and its volume is 77 cm3 then its height is ........ cm

A) 5 B) 6 C) 7-1 D) 8+2

2. The volume of cone whose height 9m is 462 m3. Then its radius is ...m

A) 5+2 B) 6+2 C) 7 D) 8-1

3. If the base radius of a cone is 3cm, height is 7 cm then the total surface area is .... cm2

A) 66 B) 132/2 C) 56 D) 55



1. Column - I column - II

i) Volume of cylinder P) rl

ii) Curved surface area of cylinder Q) r(r l)

iii) Total surface area of cylinder R) 2r h

iv) Total surface area of cone S) 2 r(r h)

T) 2 rha) i) -T, ii) S, iii) R, iv) P b) i) S, ii) T, iii) R, iv) P

c) i) R, ii) T, iii) S iv) Q d) i) R, ii) T, iii) S iv) P

III) Comprehension Type:

This section contains paragraph. Based upon each paragraph multiple choice questions have



29 VII - CLASS

to be answered. Each question has 4 choices (A) , (B) ,(C ) and (D) out of which ONLY ONE is correct.Choose the correct option.

A. The diameter of the base of a right circular cone is 8 m and its height is 3 m.Then

1. Slant height is ............ m

a) 7 m B) 5 m C) 6 m D) 9 m

2. Curved surface area is ............. m2

A) 6

627

B) 6

637

C) 6

577

D) 7

636

3. Volume of cone is ............. m3

A) 584

7B)

548

7C)

572

7D)

527

7

I) MCQ’s with Single coorect answer type :

1. Bangles of 2 mm thickness are placed one above the other to form a cylinder of curvedsurface are 572 cm2. The number of bangles placed if radius of each bangle is 1.4 cm is.......... (AS RAO 2014)

A) 325 B) 65 C) 260 D) 250

2. In case of right circular cylinder the radius of base and height are in the ratio 2:3

therefore the ratio of CSA to the TSA is .......... (NSEJS - 2013)

A) 5:3 B) 3:5 C) 2:5 D) 2:3

3. A water filter advertisement claims to provide 8 litres of water per hour. How much timedoes it take to fill four bottles of 1.5 litres each (NSEJS - 2014)

A) 45 min B) 1 hr C) 105 min D) 2 hrs

TEACHING TASK :I. 1.C 2.A 3.D 4.A 5.A 6.B

II. 1.B,D 2.A,C

III. 1.C

IV. 1.A 2.B 3.C 4.B


I. 1.C 2.A 3.C

EXPLORERS :

I. 1.B,C,D 2.A,C,D 3.A,B

II. 1.C

III. 1.B 2.A 3.C

RESEARCHERS :

1.A 2.B 3.A

RESEARCHERS ( Level - IV )

KEY


30 VII - CLASS

§§ SPHERE :

The set of all points in space which are eqisdistant from a fixed point is called a sphere

(or) A sphere is a solid generated by the rotation of a semi-circle about its diameter.The fixed

point is called the centre of the sphere and the constant distance is called its radius.

Some examples of spheres are cricket ball, globe etc..

Take a sphere and wind a string completely around the ball. Slowly remove the string from

the surface of the sphere. Start filling the circles with the string you had wound around the ball.

The string, which had completely coverd the surface area of sphere, has been used to

completely fil the area of four circles, all have same radius ofthe sphere.

If ‘r’ is the radius of sphere,

Curved surface area = 4 x area of circle

= 4 x pr2

= 4p r2 square units

Volume of sphere = 34

pr3 cubic units.

§§ HEMISPHERE :

A phone passing through the centre of a sphere divedes the sphere into two equal parts,

each of which is called a hemiphere.A sphere will have only one curved surface but a hemisphere

will have a curved surface and a base .

Consider a hemisphere of radius ‘r’ then

Curved surface area = 21

(curved surface area of shpere)

=21

[ 4p r2 ]

=2p r2 sq. unitsTotal surface area = curved surface area + Area of base

=2pr2 +pr2


31 VII - CLASS

=3pr2 sq. units

Volume of hemisphere= 21

(volume of shpere)

=

3r34

21

= 3r3

2 cu. units

§§ PYRAMID :A pyramid is a solid objcet with its bases, one end as a polygon, the other end as a vertex

and the lateral surface as triangles and the triangles meet at a common point called the vertex.The length of the perpendicular segment from the vertex to its base is called the height of

the pyramid.

The length of the perpendicular segment from the vertex to any of the sides of the regular polygon is called the start height of the right pyramid.§§ Right Pyramid :

If the base is a regualar polygon and the foot of perpendicular segment from the vertex concides with the centre of the polygon, then the solid is called a right pyramid.

Lateral surface are = 21

x perimeter of base x slant height

Total surface area = lateral surface area + Area of base

Volume of pyramid = 3

1x Area of base x height.

I) MCQ’s with single correct answr type:

1. The LSA of a right pyramid is 260cm2 whose base is a square of side 10cm. Then itsslant height is ------------cm

a) 11 b) 12 c) 13 d) 14

2. The volume of pyramid if its base is an equilateral triangle with side 6cm and height 12cm is ___________ cm3

a) 13 3 b) 36 3 c) 63 3 d) 24 33. If the base of a right pyramid is a square of side 8 cm and height is 3 cm. Then the lateral

surface area of pyramid is _____________cm2

a)48 b)80 c) 154 d) 218

4. The diameter of a spherical ball is 21cm. The amount of leather required to perpare 5such balls is _______cm2.

a) 6309 b) 6900 c) 6390 d) 6930

TEACHING TASK


32 VII - CLASS

5. If a cylinder, whose height is two -thirds of its diameter, has the same volume as a sphereof radius 4cm. Then the diameter of the base of the cylinder is_________cm

a) 8 b) 4 c) 12 d) 16

II) MCQ’s with More than one answer type :



1. A Hemispherical bowl is made of brass, 0.25cm thickess. The inner radius of hte bowl is5cm. Then the ratio of outer surface area of inner surface area is

a) 11:10 b) 21:20 c) 441:400 d) 4410:4000

2. A hemispherical dome of a building needs to be painted. The circunfernce of the base ofthe done is 17.6cm. If the cost of painting is Rs. 5 per 100 cm2. Then total cost of paintingis _________rs.

a) 4928x5 b) 2464x10 c) 26440 d) 24640

3. A hemispherical bowl has diameter 9cm. The liquid is powerd into cylindrical bottles ofdiameter 3cm and height 3cm. If a full bowl of liquid is filled into bottles, then the numberof bottles required is ____

a) 6x1.5 b) 18/2 c) 9 d) 15




i) The volume of pyramidif its base is an equilateral

triangle with side 6cm and height 12cm is ........ 3cm P) 3010

ii) The volume of a right pyramidhaving base area 215 2cm

and height 42cm is ... 3cm Q) 2940

iii) If the volume and height of a square pyramid are 8788 3cm

and 52cm resly, then the side of the base is .... cm R) 36 3

iv)The base of a right pyramid is a square with 42cm

if the height of it is 28cm then L.S.A of pyramid is ... 2cm S) 13 3

T) 18 3

a) i) T, ii) S, iii) R, iv) P b) i) S, ii) T, iii) R, iv) P

c) i) R, ii) P, iii) S, iv) Q d) i) R, ii) T, iii) S iv) P

IV) Comprehension Type :

This section contains paragraph. Based upon each paragraph multiple choice questions have to beanswered. Each question has 4 choices (A) , (B) ,(C ) and (D) out of which ONLY ONE is correct. Choosethe correct option.The internal and external radii of a hemi spherical metalic vessel are 7cm and 10.5cm

respectively. If 1 3cm of the metal weight 10g, then

1. External surface area is __________cm2


33 VII - CLASS

a) 500 b)693 c)600 d)400

2. Total surface area is _________cm2

a) 1193.5 b)2934.7 c)3725.4 d)8295.9

3. Weight of the vessel is is _________kg.

a)19.08 b) 17.07 c)15.08 d) 14.92

I) MCQ’s with Single correct answer type:

1. The volume of a pyramid whose side of a square base is 12 cm and height 15cm is__________cm3

a) 45 b) 90 c) 2160 d) 720

2. A prism and a pyramid have equal bases and height then their corresponding volumesare in the ratio of ____________

a) 1:1 b) 2:1 c) 3:1 d) 1:2

3. The volume of a sphere whose diameter 14cm is __________cm3

a) 5226.6 b) 52.66 c) 522 d) 522.66

4. The total surface area of a hemisphere whose radius 10cm is ________cm2

a) 941.85 b) 942.85 c) 941 d) 942

5. If a spherical ball of diameter 21cm. is melted and recast into cubes, each of side 1cm.Then the no.of cubes thus formed are__________

a) 4451 b) 4751 c) 4851 d) 4351

6. The cost of painting the outer surface of a closed box which is 60cm long, 40cm broadand 30cm high at the rate of 50 paise per 20 cm2 is ___________is

a) 200 b) 240 c) 270 d) 320

7. If 42 cubes of side 3cms each can be cut from wooden block in the form of a cuboidwhose length breadth and height are 21cm, 9cm and 8cm respectively the volume ofwood wasted is _______cm3.

a) 376 b) 388 c) 378 d) 387

8. A cube of edge 4 cm is painted externally it is then cut into one cm cubes., How many ofthose do not have red paint on any face?

a) 4 b) 8 c) 56 d) 16

9. If the total surface area of a cone is 21

1137

m and its height is 3cm. Then the diameter of

the base of a cone is ............ m

A) 6 B) 7 C) 8 D) 9

10. A right triangle, whose sides are 15cm and 20cm is made to revolve about its hypotenuse.Then the surface area of the double cone so formed is ......... cm2

A) 1023 B) 1032 C) 1320 D) 1230


1. How many spherical lead shots each 4.2cm in diameter can be obtained from a rectanglularsolid of lead with dimensions 66cm, 42cm, 21cm?


LEARNER’S TASK



34 VII - CLASS

2. The largest sphere is carved out of a cube of side 7cm. Find the volume of the sphere.

3. If the diameter of a sphere is deereased by 25% then by percent its curved

surface area decrease?

4. The interanl and external radii of a hemi spherical metallic vessel are 7cm and 10.5 cmrespectively. If 1 cm3 of the metal weigth 10gms then find the total weight of the vessel.

5. A toy is in the form of acone mounted an a hemisphere of diameter 7cm. The total heightof the toy is 15.5cm Find the volume and the total surface area of the toy.

6. A metallic sphere of radius 10.5cm is melted and then recast into smaller canes, each ofradius 3.5cm and height 3cm. Find how many cones are obtained.

I) MCQ’s with More than one answer type :

This section contains multiple choice questions. Each question has 4 choices (A), (B),

(C),(D), out of which ONE or MORE is correct. Choose the correct options

1. The ratio of radii of two sphere is 2:3 then the ratio of their volumes is ________

a) 9:4 b) 16:54 c) 27:8 d) 8:27

2. The length of equator of the globe is 44cm. Then its surface area is ____cm2.

a) 616 b) 308x2 c) 717 d) 1800

3. The number of liters of mile can a hemispherical bowl of diameter 10.5cm can hold is____

a) 0.03 b) 0.303 c) 303 d) 303/1000

4. If the surface area of a shpere is 154cm2 then its radius is _____cm

a) 7 b) 4

7c)

2

7d) 3.5




i) surface area of a sphere p) 2r3

ii) volume of hemisphere q) 3r34

iii) volume of a sphere r) 4pr2

iv) surface area of hemisphere s) 3r31

t) 3r3

2

a) i-r, ii-t, iii-q, iv-p b) i-t,ii-q, iii-p, iv-s

c) i-t, ii-q, iii-s, iv-p d) i-t, ii-r, iii-q, iv-p

III) Comprehension Type :




35 VII - CLASS

. The base of a right pyramid is an equilateral triangle of side 6m and every slant edge 5mlong. Then

1. lateral surface area is __________cm2

a) )34(9 b) 39 c)36 d) 345

2. Total surface area is _________cm2

a) )34(9 b) 39 c)36 d) 345

3. Volume of athe pyramid is _________cm3

a) 39 b) 393 c) 339 d) 39

TEACHING TASK :I. 1.C 2.B 3.B 4.D 5.A

II. 1.B,C,D 2.A,B,D 3.A,B,C

III. 1.C IV. 1.B 2.A 3.B


I) 1.B 2.B 3.D 4.B 5.C 6.C 7.C 8.D 9.C10.C

EXPLORERS :

I) 1.B,D 2.A,B 3.B,D 4.C,D

II) 1.A III) 1.C 2.A 3.B

KEY

mensuration - logical class

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