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MENSURATION
H ere, we cover three areas
1. Perimeter 2. Area
3. Volume
To find these things we have to
remember the formulae of plane
figures and solid figures. Try to
understand the concept behind
reaching the formula.
1. The area of a rectangular plate is 236.25
sq.cm. If the length of the plate is 17.5 cm,
find the perimeter of the plate.
1) 65 cm
2) 52 cm
3) 62 cm
4) 67 cm
5) None of these
2. The breadth of a rectangular plot is 75% of
its length. If the perimeter of the plot be
1050 m, what is its area?
1) 12500
2) 57500
3) 62500
4) 67500
5) None of these
3. In measuring the sides of a rectangle, one
side is taken 10% in excess, and the other
5% in deficit. What is the change in its area
as a percentage?
1) 1.5%
2) 4.5%
3) 2.5%
4) 5%
5) None of these
4. How many tiles of 20 cm length and 10 cm
width are required to pave the floor of a room
8 m long and 5 m wide?
1) 2100
2) 1500
3) 2500
4) 2000
5) None of these
5. The circumference of two circles is 88 metre
and 132 metre respectively. What is the difference between the area of the larger circle
and the smaller circle?
1) 440 sq.m
2) 510 sq.m 3) 770 sq.m
4) 225 sq.m
5) None of these
6. A solid metallic spherical ball of radius 7 cm
is melted down and recast into small cones.
If the diameter of the base of the cone is
14 cm and the height is 2 cm, find the number of such cones can be made.
1) 14
2) 28
3) 7
4) 12
5) None of these
7. A hall is 10metres long and 8metres wide.
What will be the cost of carpeting the room if
0.5 metres of space is left around the room?
The rate of 0.25 metre wide carpet is Rs.20
per metre.
1) Rs.1560
2) Rs.2520
3) Rs.1875
4) Rs.5040
5) None of these
8. A square field has an area of 50625 sq. m.
Find the cost of fencing around it at Rs. 15
per metre.
1) Rs.12500
2) Rs.17500
3) Rs.13500
4) Rs.16250
5) None of these
9. Cost of fencing a circular plot at the rate of
Rs.12 per metre is Rs.2,640. What will be
the cost of carpeting the floor of the plot at
the rate of Rs.85 per square metre?
1) Rs.3,25,250
2) Rs.3,22,350
3) Rs.3,27,250
4) Rs.3,25,750
5) None of these
10. A circle and a rectangle have the same
perimeter. The sides of the rectangle are
18 cm and 26 cm. What will be the area of
the circle?
1) 88 cm2
2) 1250 cm2
2
3) 154 cm
4) 128 cm2
5) None of these
11. A circular garden has a 7 m wide road
around the border. Find the cost of leveling
the road at Rs.5 per square meter if the
radius of the inner circle is 21 m?
1) Rs.3490
2) Rs.5390
3) Rs.2930
4) Rs.5695
5) None of these
1
Mensuration means measurement. It is being done in our life in
many situations. For example, Length of cloth we need for
stitching, the area of a wall which is being painted, perimeter of
a rectangular plot to be fenced, quantity of water needed to fill
the tank etc. For these kinds of activities, we are doing
measurements for further needs.
What will be the area of the circle?
12. The lateral surface area of a cylinder is
EXPLANATIONS
thrice the area of its base. Find the ratio of
236.251
its height and the base radius?
1. Width of the plate = ⎯⎯⎯⎯⎯ = 13.5 cm
17.5
1) 2:3
2) 2:5
3) 3:2
Perimeter = 2 (length + width)
4) 4:9
5) None of these
⇒ 2 (17.5 + 13.5) = 62 cm.
13. Two cubes have their volumes in the ratio
2.
Let
the
length
of the plot be x m
8:27. Find the ratio of their surface areas.
75
3x
1) 2:3
2) 4:9
3) 16:81
∴ Breadth = ⎯⎯ × x = ⎯ m
100
4
4) 8:9
5) None of these
Perimeter
=
1050.
14. A horse is tied with a 14 m long rope. How
3x
∴ 1050 = 2 x + ⎯
much ground will it be able to graze?
4
1) 125 Sq.m
∴ x = 300 m.
2) 275 Sq.m
∴ Area of the plot
3) 625 Sq.m
3 × 300
= 300 × ⎯⎯⎯
4) 675 Sq.m
4
2
5) None of these
= 67500 m
Quantitative Aptitude
15. Find the length of the
3. Let the length be 10 and
longest stick that can be
breadth also be 10.
placed in a room of 12 m long, 9
∴ Area = 10 × 10 = 100
m broad and 8 m high.
New length (10% excess) = 11,
1) 21 m
2) 29 m
New breadth (5% deficit) = 9.5
3) 20 m
4) 17 m
New area = 11 × 9.5 = 104.5
∴ Change in area = 104.5 − 100
5) None of these
= 4.5
16. A field is 375 m long and 40 m
4.5
broad. A tank 30 m long, 20 m
G.S. Giridhar
Percentage change = ⎯⎯ × 100
broad and 12 m deep is dug in
100
the field and earth taken out of it and is
= 4.5%
spread equally over the field. How much is
4. Area of the floor of the room = 800 × 500
the level of field raised?
= 400000 cm2
1) 48 cm
2) 0.5 cm 3) 25 cm
Area of the Tile = 20 ×10 cm2
4) 50 cm
5) None of these
400000
∴ Number of Tiles required = ⎯⎯⎯⎯
17. A rectangular field of 60-metre length and
20 × 10
40 metres wide is to be surrounded by a
= 2000
road 5 meter wide. If the cost of making 1
5. Let the radius of larger circle be 'R' and
square meter road is Rs.500, what would
smaller circle be 'r'
be the cost of the entire road?
22
∴ 2 π R = 132 ⇒ 2 × ⎯ × R = 132 ⇒ R = 21
1) Rs.6,25,000
2) Rs.5,55,000
7
22
3) Rs.5,50,000
4) Rs.5,67,500
2πr = 88 ⇒ 2 × ⎯ × r = 88 ⇒ r = 14
7
5) None of these
Difference in the areas = π R2 − π r2
18. A room is 7.5 m long, 5.5 m broad and 5 m
= π R2− r2
high. What will be the expenditure in cover22
22
ing the walls by paper 40 cm broad at the
⇒ ⎯ (212 − 142) ⇒ ⎯ (21 + 14) (21 − 14)
7
7
rate of 75 paise per meter?
22
⇒ ⎯ × 35 × 7 = 770 sq.metres
1) Rs.245.25
2) Rs.243.75
(
)
(
)
7
3) Rs.275.50
4) Rs.275.45
5) None of these
19. The length and breadth of a rectangle
increased by 20% and 10% respectively.
How much percent will its area be
increased?
1) 25%
2) 30%
3) 40%
4) 32%
5) None of these
20. 7 cm radius and 28 cm height solid metallic
cylinder is melted and recast into small
spherical balls of 7cm radius. Find the number of cubes that can be made.
1) 12
2) 5
3) 3
4) 6
5) None of these
ANSWERS
1-3
2-4
3-2
4-4
5-3
6-1
7-4
8-3
9-3
10-5
11-2
12-3
13-2
14-5
15-4
16-4
17-3
18-2
19-4
20-3
(Writer − Director, RACE Institute, Hyderabad)
Mail your comments and suggestions to
[email protected]
4
6. Volume of sphere = b ⎯ πr3
3
1 2
Volume of cone = ⎯ πr h
3
Let the number of cones be 'x'
4
1
∴ ⎯ πr3 = ⎯ πr2h × x
3
3
4
1
⇒ ⎯ π × 73 = ⎯ π × 72 × 2 × x
3
3
⇒ x = 14
7. Length of the carpet = 10 − (0.5 + 0.5) = 9 m
Width of the carpet = 8 − (0.5 + 0.5) = 7 m
Area of carpet = 9 × 7 = 63
63
Required length of carpet = ⎯⎯ = 252 m
0.25
∴Cost of carpet = 252 × Rs.20 = Rs.5,040.
⎯⎯⎯
8. Side of the square field = √ 50625 = 225 m
Length of the wire required = 4 × 225
= 900 m
∴Total cost = Rs.15 × 900 = Rs.13,500
9. Circumference of the circular plot
2640
= ⎯⎯⎯ = 220 m
12
22
2πr = 220 ⇒ 2× ⎯ × r = 220 ⇒ r = 35 m
7
Area of the plot
22
= πr2 = ⎯ × 35 × 35 = 3850 sq.m
7
∴ Cost of carpeting = 3850 × Rs.85
= Rs.3,27,250
10. Perimeter of rectangle = 2(18 + 26) = 88 cm
Circumference of circle 2πr = 88
22
⇒ 2 × ⎯ × r = 88 ⇒ r = 14 cm
7
22
Area of circle = πr2 = ⎯ × 14 × 14
7
= 616 cm2
11. Radius of outer circle = 21 + 7 = 28 m
Road area = Area of outside circle − Area of
inside circle
Road area = π(28)2 − π(21)2 = 1078 sq.m
∴Cost = 1078 × Rs.5 = Rs.5,390.
12. Lateral surface area of cylinder = 2πrh
Base area of cylinder = πr2
∴ 2πrh = 3πr2
⇒ h: r = 3 : 2
13. Let the sides of two cubes be a1 and a2
a13 : a 23 = 8 : 27 ⇒ a1: a2 = 2 : 3
∴ Surface areas ratio = 6a12 : 6a22
= (2)2 : (3)2 = 4 : 9
14. Length of the rope will be the radius of the
circle = 14 m
Area of the circle
22
= πr2 = ⎯ × 14 × 14 = 616 sq.m
7
∴ Area of the ground that can be grazed
= 616 sq.m.
15. Length of the stick = length of the diagonal
of the room
⎯⎯⎯⎯⎯
⎯⎯
√ 122 + 92 + 82 = √ 289
= 17 m
16. Quantity of the earth taken from the tank
= 30 × 20 × 12 cu.m
Area of the field for the earth to be spreaded
= (375 × 40) − (30 × 20) = 14400
∴ The rise in the level of the field
30 × 20 × 12
= ⎯⎯⎯⎯⎯ × 100 = 50 cm
14400
17. Area of the road
= Area of outside rectangle − Area of inside
rectangle
= 70 × 50 − 60 × 40
= 3500 − 2400 = 1100 m2
∴ The cost of making 1100 m2
= 500 × 1100 = Rs.5,50,000
18. Area of four walls = 2 × 5 (7.5 + 5.5)
= 130 m2
Area of required paper = 130 m2
Breadth of the paper = 40 cm = 0.4 m
130
∴ Length of the paper = ⎯⎯ = 325 m
0.4
∴ Cost of paper at 75 paise per meter
= 325 × 0.75 = Rs.243.75
19. Let the length and breadth be 10 and 10
Area = 10 × 10 = 100
New length = 12 (20% increase)
New breadth = 11 (10% increase)
New area = 12 × 11 = 132
∴ Increment in area = 132 − 100 = 32%.
20. Let the number of spherical balls be 'x'
Volume of cylinder = x × Volume of sphere
4
π × 72 × 28 = x × ⎯ × π × 73
3
x=3