Explanation / Important formulas:
Area refers to the space inside a twodimensional object, like a square or an octagon, while volume refers to the space that a threedimensional object takes up, or the internal capacity of that object. Units of area include square feet or square meters, while units of volume include cubic feet or cubic meters.
SPHERE
Let the radius of the sphere be r. Then,
 Volume of the sphere = (4/3 πr^{3}) cubic units
 Surface area of the sphere = (4πr^{2}) sq. units
CYLINDER
Let radius of base = r and Height (or length) = h. Then,
 Volume of the cylinder = (πr^{2}h) cubic units
 Curved surface area of the cylinder = (2πrh) sq. units
 Total surface area of the cylinder = 2πr(h + r) sq. units
CUBE
Let each edge of a cube be of length a. Then,
 Volume of the cube = a^{3} cubic units
 Surface area of the cube = 6a^{2} sq. units
 Diagonal = 3a units
CUBOID
Let length = l breadth = b and height = h units. Then,
 Volume of cuboids = (l x b x h) cubic units
 Surface area of cuboids = 2(lb + bh + lh) sq. units
 Diagonal = l^{2} + b^{2} + h^{2} units
CONE
Let radius of base = r and Height = h. Then,
 Slant height, l = h^{2} + r^{2} units
 Volume of the cone = (1/3 π r^{2}h)cubic units
 Curved surface area = (πrl) sq. units
 Total surface area = (πrl + πr^{2}) sq. units
HEMISPHERE
Let the radius of a hemisphere be r. Then,
 Volume of the hemisphere = (2/3 πr^{3}) cubic units
 Curved surface area of the hemisphere = (2πr^{2}) sq. units
 Total surface area of the hemisphere = (3πr^{2}) sq. units
 1 litre = 1000 cm^{3}
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Test  Area and volume
Area and volume  Question and Answers
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Question 1

A

2500

B

500

C

100

D

50

Question 2

A

8

B

16

C

32

D

64

No. of blocks needed = (8*8*8) / (2*4*8) = 8
Question 3

A

12 m

B

10 m

C

8 m

D

6 m

= 3cube + 3cube + 5cube = 216 m cube
Side of the new cube = 6m
Question 4

A

108

B

54

C

27

D

6

Number of cubes = 216/8 = 27
Question 5

A

150

B

50

C

15

D

6

= Volume of the cuboid / volume of a cube
= (10*15*5) / (5*5*5) = 6
Question 6

A

1

B

2

C

3

D

6

= 27 mcube
Side of the smaller cube = 3m
Question 7

A

10 m cube

B

20 m cube

C

50 m cube

D

100 m cube

Volume of the cuboid = 550/11 = 50 m cube
Question 8

A

600 m

B

300 m

C

150 m

D

115 m

= 39600 / π d
= 600
Question 9

A

100 cm

B

50 cm

C

25 cm

D

20 cm

53 = 0.2*a square
a = 25 cm
Question 10

A

0.88 hrs

B

1.76 hrs

C

50.4 hrs

D

61.6 hrs

= 440 m
Time to complete 20 rounds = 20*440/5000
= 1.76 hrs
Question 11

A

12.5 cm square

B

25 cm square

C

50 cm square

D

100 cm square

√2 a = 2r = 10 cm
a square = 50 cm square
Question 12

A

10 m

B

10 m

C

7.5 m

D

4.8 m

H = 4.8 m
Question 13

A

30 cm square

B

36 cm square

C

216 cm square

D

180 cm square

X = 6
Area = 12*18 = 216 cm square
Question 14

A

54 cm

B

56 cm

C

58 cm

D

60 cm

2(l+b) = 54
Solving the two equations, l = 15cm, b = 12cm
New perimeter = 2*(18+10) = 56cm
Question 15

A

480 liters

B

240 liters

C

240000 liters

D

480000 liters

= 480 cubic meters
Converting cubic meter to liters,
Capacity of the tank = 480*1000
= 480,000 liters
Question 16

A

6 units

B

36 units

C

12 units

D

10 units

Volume of a cube = a cube cubic units
Surface area = 6 a square Square units
a cube = a square
a = 6 units
Question 17

A

170m square

B

136m square

C

15m square

D

120m square

I^{2} + b^{2} = 289
I^{2} = 289 – 64 = 255
l = 15m
Area = 15*8 = 120 m square
Question 18

A

6 mins

B

30 mins

C

30 mins

D

5 mins

= 100 √ 2 m
Diagonal of the field = √ 2 a
= 200 m
Time taken to cross 200 m
= 0.2/4
= 3 minutes
Question 19

A

3:1

B

1:3

C

1:9

D

27:1

= 54 cm square
Surface area of a 1 cm cube = 6*12 = 6 cm square
No. of smaller cubes formed = Volume of large cube /Volume of smaller cube
= 3 cube/ 1 cube
= 27
Ratio = 54/ (27*6)
= 1:3
Question 20

A

2

B

6

C

12

D

24

Area which can be grazed by the cow = Π r^{2}
= 616 sq.ft
Area grazed in a day = 50 sq.ft
Time taken to graze 616 sq.ft = 616/50
= 12.13 days
Question 21

A

6.25%

B

12.5%

C

12.5%

D

22%

= 2*4*5.5 = 44 cm square
Volume of the cylinder = Π r^{2} h cubic units
= (22/7)*(4*4)*7 cm square
= 352 cm square
% of space occupied by the block = (44/352)*100
= 12.5%
Question 22

A

150%

B

600%

C

500%

D

400%

No. of smaller cubes formed = Volume of large cube / Volume of smaller cube
= 53 / 13
= 125
Sum of surface areas of smaller cubes = 125*6 = 750 m square
Increase in surface area = 600 m square
% increase = (600/150)*100
= 400%
Question 23

A

74 cm cube

B

218 cm cube

C

400 cm cube

D

468 cm cube

Volume of the second cube = 73 = 343 cm cube
Volume of the new cube = 125+343 = 468 cm cube
Question 24

A

lb +x(l+b)+x square

B

lb +x(l + b)

C

x (l + b)+x square

D

4x

= lb+x (l+b) + x square
Increase in area = x (l+b) + x square
Question 25

A

450m X 300m

B

150m X 100m

C

480m X 320m

D

100m X 100m

In 60 mins, he covers 24 km.
In 4 mins, he covers
24000/15 = 1600 m
Perimeter of the park = 1600 m
L: b = 3:2
2(l+b) = 1600
2*(3x+2x) = 1600
X = 160
l = 480m, b = 320m
Question 26

A

45 feet

B

63 feet

C

275 feet

D

315 feet

X*7 = (x10)*9
7x= 9x90
X= 45
Distance = 45*7 = 315 feet
Question 27

A

30%

B

18.75%

C

15%

D

13.6%

Area of the 4 walls = 2h (l+b) = 2x*5x = 10x^{2}
New length, breadth and height = 6x, x, (x/2)
New area = 2*(x/2)*(7x) = 7x^{2}
Decrease in area = 3x^{2}
% decrease = (3x^{2}/10x^{2})*100
= 30%
Question 28

A

5.5

B

4.5

C

7.5

D

6.5

If the volume is reduced by 19%, new volume = 81% of 60
= 48.6
Let the length and breadth be reduced to x%
((x/100*6)*(x/100)*5)*2 = 48.6
6x^{2} = 48600
x^{2} = 8100
x= 90
New width = 90% of 5 = 4.5
Question 29

A

6K

B

8K

C

12K

D

7K

2l*4w = 8lw
8lw – lw = 7lw = 7K
Question 30

The length of the side of a square is represented by x+2. The length of the side of an equilateral triangle is 2x. If the square and the equilateral triangle have equal perimeter, then the value of x is
A

3

B

4

C

5

D

6

Perimeter of the triangle = 6x
4x+8 = 6x
X = 4
Question 31

A

16%

B

44%

C

36%

D

40%

New radius = 0.8r
New area = 0.64 √r^{2}
Decrease in area = 0.36 √r^{2}
36% decrease
Question 32

A

x^{2}/9

B

x^{2}/8

C

x^{2}/4

D

x^{2}

X feet of fencing covers three sides
x = 2l+b
b = x2l
Area of the yard = lb square yard
= l(x2l)
=  x2  square
To get the maximum possible value for the area, d (area) / dl = 0
x  4l = 0
l = x/4
The maximum value for each occurs when l = x/4
b = x = 21 = x/2
Area = (x/4) (x/2)
= x^{2}/ 8
Question 33

A

24

B

120

C

240

D

450

= 10*9*5
= 450
Question 34

A

100%

B

200%

C

700%

D

800%

R = 2r
V = 8*(4/3) √r^{3}
The volume increase by 800%
Question 35

What is the percentage increase in area when a triangle is cloned (so that we have two triangles in total) and the resulting two triangles are joined on their bases form a parallelogram?
A

100%

B

150%

C

300%

D

120%

Hence, the area is increased by 100%
Question 36

A

2424 cm^{2}

B

2446 cm^{2}

C

2464 cm^{2}

D

2484 cm^{2}

R = 14
CSA of a sphere = 4 √r^{2} = 2464 cm^{2}
Question 37

A

100%

B

200%

C

300%

D

400%

If the height is doubled, the volume is also doubled.
Hence, the volume increases by 100%
Question 38

A

Rs. 1050

B

Rs. 1400

C

Rs. 3150

D

Rs. 4200

Area of the wall 4 walls = 2h (l+b)
If the length, breadth and height are 3l, 3b and 3h,
Area of the 4 walls = 18h (l+b)
Cost = 350*9 = Rs. 3150
Question 39

^{ }
A

10 cm

B

15 cm

C

18 cm

D

24 cm

h = 3h = 15 cm
Question 40

50 men took a dip in water tank 40 m long and 20 m broad on a religious day. If the average displacements of water by a man are 4m^{3}, then the rise in the water level in the tank will be?
A

20 cm

B

25 cm

C

35 cm

D

50 cm

40*20*h = 200
H = ¼ m = 25 cm
Question 41

^{ }
A

2h

B

4h

C

2h/3

D

h

1/3 √r^{2}h +  r^{2}H = 3*(1/3 √r^{2} h)
h/3 + H = h
H = 2h/3
Question 42

A

1:5

B

1:25

C

1:125

D

1:625

= 1 : 25
Question 43

A

84

B

0.84

C

8.4

D

0.084

= 0.84
Question 44

A

196 cm^{2}

B

784 cm^{2}

C

1176 cm^{2}

D

588 cm^{2}

a = 14
62 = 1176 cm^{2}
Question 45

A

36

B

216

C

218

D

432

= 216
Question 46

A

56 kg

B

36 kg

C

48 kg

D

27 kg

= 36
Question 47

A

32 m^{3}

B

36 m^{3}

C

40 m^{3}

D

44 m^{3}

Volume of earth = √r^{2} h
= 44 m^{3}
Question 48

A

1:4

B

1:16

C

4:1

D

16:1

Surface area is proportional to the square of the radius
A1: A2 = 16: 1
Question 49

A

1300 m^{3}

B

1331 m^{3}

C

1452 m^{3}

D

1542 m^{3}

a = 11
Volume = a^{3} = 1331
Question 50

A

is doubled

B

increases 6 times

C

increases 4 times

D

increases 8 times

If the edge is doubled, the volume becomes 8 times
Question 51

A

120 liters

B

12000 liters

C

1200 liters

D

120000 liters

= 120000 liters
Question 52

A

3 cm

B

4 cm

C

6 cm

D

8 cm

R = 6 cm
Question 53

A

4:25

B

4:15

C

3:25

D

3:15

Ratio = 4:25
Question 54

A

22 cm

B

24 cm

C

26 cm

D

28 cm

4a2 = 32
a1 = 10
a2 = 8
a square = a1 square – a2 square = 36
a = 6
4a = 24
Question 55

A

18 m

B

20 m

C

22 m

D

25 m

Lb = 460
1.15 b^{2}= 460
b^{2} = 400
b = 20
Question 56

A

152600 m^{2}

B

153500 m^{2}

C

153600 m^{2}

D

153800 m^{2}

= 8*12000/60
= 1600
2 (l+b) = 1600
l+b = 800
l:b = 3:2
Solving, l = 480 b = 320
Area = lb = 153600
Question 57

A

42 m^{2}

B

49 m^{2}

C

52 m^{2}

D

64 m^{2}

= 2*1.25(4+6) + 6*4
= 49
Question 58

A

800

B

125

C

400

D

8000

= 8000
Question 59

A

6 m

B

4 m

C

2 m

D

3 m

Area of the lawn = 2109 m^{2}
Area of the cross roads = (2400  2109) m^{2} = 291 m^{2}
Let the width of the road be x meters, Then,
60x+40x pow(x,2) = 291
x^{2} – 100x+291 = 0
(x  97)(x 3) = 0
x = 3
Question 60

A

156300

B

153060

C

153600

D

153006

Let length = 3x meters and breadth = 2x meters
Then, 2(3x +2x) = 1600 or x = 160.
Length = 480 m and breadth = 320 m
Area = (480 x 320)m^{2} = 153600 m^{2}
Question 61

A

4.04%

B

4.40%

C

4.35%

D

4.53%

(A2 – A1) = [pow(102,2) – pow(100,2)]
= (102 +100) x (102  100)
= 404 cm^{2}
Percentage error = 404/(100 x 100) x 100% = 4.04%
Question 62

A

24%

B

30%

C

82%

D

28%

Decrease in area
= xy – [(80/100) x] x (90/100) y
= xy – (18/25)xy = (7/25)xy
Decrease% = [(7/25)xy] x [1/(xy)] x100% = 28%
Question 63

A

148

B

814

C

841

D

418

Area of each tile = (41 x 41) cm^{2}
Required number of titles = (1517 x 902) / (41 x 41) = 814
Question 64

A

5220

B

5202

C

2502

D

2520

Solving the two equation, we get l = 63 and b = 40
Area = (l x b) = (63 x 40)m^{2} = 2520 m^{2}
Question 65

A

50%

B

40%

C

45%

D

35%

Original area = xy
New length = x/2
New breadth = 3y
New area = (x/2) x 3y = (3/2)xy
Increase % = [(1/2(xy))] x (1/xy) x 100% = 50%
Question 66

A

50 m

B

60 m

C

45 m

D

75 m

Then, length = (x+20) meters
Perimeter = (5300/26.50) m = 200 m
2[(x+20) +x] = 200
2x+20 = 100
2x = 80
X = 40
Hence, length = x + 20 = 60 m
Question 67

A

88 ft

B

77 ft

C

88.5 ft

D

76 ft

So, b = 34 ft
Length of fencing = (l+2b) = (20+68) ft = 88 ft.
Question 68

A

Rs. 555

B

Rs. 855

C

Rs. 585

D

Rs. 558

= {[2(25+12) x 6] + (25 x 12)} m^{2}
= (444+300) m^{2}n
= 744m^{2}
Cost of plastering = Rs. 744 x (75/100) = Rs. 558
Question 69

A

25 cm

B

30 cm

C

35 cm

D

40 cm

Let one diagonal be d1 and d2
So as per question
(1/2)*d1*d2 = 150
(1/2)*10*d2 = 150 d2 = 150/5 = 30
Question 70

A

350 m

B

300 m

C

200 m

D

250 m

Let the altitude of triangle is h1 and of parallelogram is h2 (which is equal to 100m), then
Area of triangle = (1/2)*b*h1
Area of rectangle = b*h2
As per question (1/2)*b*h1 = b*h2
(1/2)*b*h1 = b*100
h1 = 100*2 = 200m
Question 71

A

1:2

B

2:1

C

1:4

D

4:1

Area of rectangle = l*b
Area of triangle = (1/2) l*b
Ratio = l*b : (1/2) l*b = 1:1/2 = 2:1
Question 72

A

2a

B

3a

C

4a

D

5a

= (1/2)*a*h
= (1/2)*a*h = a2 => h= 2a
Question 73

A

20 cm

B

18 cm

C

19 cm

D

16 cm

= 90 cm^{2}
Area of second triangle = 2*A1 = 180 cm^{2}
(1/2)*20* height = 180
Height = 18 cm
Question 74

A

9 cm

B

6 cm

C

7 cm

D

8 cm

(x+2)2 – x2 = 32 x2 + 4x+4 – x2 = 32
4x = 28 x = 7cm
Question 75

A

841

B

418

C

481

D

814

Length of largest tile = Hcf of (1517 cm and 902 cm)
= 41 cm
Required number of titles = Area of floor/ Area of tile
= (1517 x 902) / (41 x 41) = 814
Question 76

A

20000

B

30000

C

25000

D

22000

Question 77

A

Rs.45

B

Rs.78

C

Rs.44

D

Rs.40

176 * ¼ = Rs.44
Question 78

A

50%

B

60%

C

64%

D

84%

R = 5 p*r*r = 25
25 p – 16 p
100 ? => 64%
Question 79

A

1120

B

1220

C

1210

D

1250

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