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.
Here, 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.
Ans. 62 cm
Explanation :
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?
Ans. 67500
Explanation :
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?
Ans. 4.5%
Explanation :
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?
Ans. 2000
Explanation :
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?
Ans. 770 sq.m
Explanation :
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.
Ans. 14
Explanation :
7. A hall is 10 metres long and 8 metres 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.
Ans. Rs.5040
Explanation :
8. A square field has an area of 50625 sq. m. Find the cost of fencing around it at Rs.15 per metre.
Ans. Rs.13500
Explanation :
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?
Ans. Rs.3,27,250
Explanation :
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 3) 154 cm2 4) 128 cm2 5) None of these
Ans. None of these
Explanation :
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?
Ans. Rs.5390
Explanation :
12. The lateral surface area of a cylinder is thrice the area of its base. Find the ratio of its height and the base radius?
Ans. 3 : 2
Explanation :
13. Two cubes have their volumes in the ratio 8 : 27. Find the ratio of their surface areas.
Ans. 4 : 9
Explanation :
14. A horse is tied with a 14 m long rope. How much ground will it be able to graze?
1) 125 Sq.m 2) 275 Sq.m 3) 625 Sq.m 4) 675 Sq.m 5) None of these
Ans. None of these
Explanation :
15. Find the length of the longest stick that can be placed in a room of 12 m long, 9 m broad and 8 m high.
Ans. 17 m
Explanation :
16. A field is 375 m long and 40 m broad. A tank 30 m long, 20 m broad and 12 m deep is dug in the field and earth taken out of it and is spread equally over the field. How much is the level of field raised?
Ans. 50 cm
Explanation :
17. A rectangular field of 60-metre length and 40 metres wide is to be surrounded by a road 5 meter wide. If the cost of making 1 square meter road is Rs.500, what would be the cost of the entire road?
Ans. Rs.5,50,000
Explanation :
18. A room is 7.5 m long, 5.5 m broad and 5 m high. What will be the expenditure in covering the walls by paper 40 cm broad at the rate of 75 paise per meter?
Ans. Rs.243.75
Explanation :
19. The length and breadth of a rectangle increased by 20% and 10% respectively. How much percent will its area be increased?
Ans. 32%
Explanation :
20. 7 cm radius and 28 cm height solid metallic cylinder is melted and recast into small spherical balls of 7 cm radius. Find the number of cubes that can be made.
Ans. 3
Explanation :
Mensuration - 2D (Plane Figur es)
1. The length of a rectangular field is four times its width. If the perimeter of the field is 30 m, find the area of the field?
a) 108 m2 b) 27 m2 c) 9 m2 d) 36 m2
2. A circular wire of radius 42 cm is cut and bent in the form of a rectangle whose sides are in the ratio 6 : 5. Find the smaller side of the rectangle?
a) 72 cm b) 120 cm c) 60 cm d) 144 cm
3. The radius of a circle is 14 cm. Find the area of the sector with central angle 36°?
a) 61.6 cm2 b) 30.8 cm2 c) 15.4 cm2 d) 308 cm2
4. In the given figure, if AB = 4 cm and BD = 4√3 cm, then the area of the shaded region will be?
a) 48 π cm2 b) 24 π cm2 c) 16 π cm2 d) 12 π cm2
5. A rectangular garden has an area of 2000 m2 and its length and breadth are in the respective ratio 5 : 4. A road of uniform width runs inside the garden around the perimeter and has an area of 344 m2. The width of the road is......
a) 3 m b) 3.5 m c) 4 m d) 2 m
6. If the perimeters of a rectangle and a square are equal and the ratio of two adjacent sides of the rectangle is 1 : 2 then the ratio of area of the rectangle and that of the square is?
a) 1 : 1 b) 1 : 2 c) 2 : 3 d) 8 : 9
7. If a wire is bent into the shape of a square, then the area of the square so formed is 81 cm2. When the wire is rebent into a semicircular shape, then the area (in cm2) of the semicircle will be (Take π = 22/7)
a) 22 b) 44 c) 77 d) 154
8. At each corner of a triangular field of sides 26 m, 28 m and 30 m, a cow is tethered by a rope of length 7 m. The ungrazed area (in m2) by the cow is.......
a) 336 b) 259 c) 154 d) 77
9. How many rotations will a wheel of a car make in a journey of 88 km, if the radius of the wheel is 56 cm?
a) 60000 b) 50000 c) 30000 d) 25000
10. A cow is tied by a rope to one of the vertices of a square of side 14 m. The length of the rope is 7 m. What percentage of the field is grazed by the cow?
a) 15% b) 25% c) 18% d) 19.6%
11. A horse is tethered by a rope. What is the length of the rope so that the horse can graze over an area of 154 m2?
a) 2 m b) 7 m c) 14 m d) 49 m
12. A circular grass lawn of 28 m radius, has a path of 14 m wide running around it on the outside. Find the area of the path?
a) 3920 m2 b) 3080 m2 c) 3980 m2 d) 8030 m2
13. The area of a circular plot is 154 m2. Find the cost of fencing it at the rate of Rs. 2.75 per metre?
a) Rs.56 b) Rs.100 c) Rs.121 d) Rs.423.50
14. A wire bent in the form of a square, encloses an area of 484 cm2. If the same wire is bent to form a circle, then the area enclosed will be
a) 1232 cm2 b) 616 cm2 c) 2464 cm2 d) 4312 cm2
15. A copper wire is bent in the form of an equilateral triangle and has area 121√3 cm2. If the same wire is bent into the form of a circle, the area (in cm2) enclosed by the wire is....... (Take π = 22/7)
a) 364.5 b) 693.5 c) 346.5 d) 639.5
16. The area of a rhombus is 150 cm2. The length of one of its diagonals is 10 cm. The length of the other diagonal is........
a) 25 cm b) 30 cm c) 35 cm d) 40 cm
17. The area of the ring between two concentric circles, with circumference 88 cm and 132 cm respectively is......
a) 780 cm2 b) 770 cm2 c) 715 cm2 d) 660 cm2
Key With Explanations
1-d; 2(l + b) = 30
⇒ 2(4b + b) = 30
⇒ 10b = 30 ⇒ b = 3 m and l = 12 m
So, area of the field = l × b = 3 × 12 = 36 m2.
2-c; Perimeter of the rectangle = Circumference of the circular wire
= 2 × 22/7 42 cm = 264 cm.
Let the dimensions of the rectangle be 6x and 5x respectively.
∴ 2 × (6x + 5x) = 264 ⇒ x = 12
∴ Smaller side = 5x = 60 cm
3-a; Radius of the circle = 14 cm
Angle of the sector = 36°
Area of the sector = θ/360° × π r2
∴ Area of the sector
= 36/360° × 22/7 × 14 × 14 = 61.6 cm2
4-d; ∠ADC is a right angle (Angle in a semicircle)
So, BD2 = AB × BC ⇒ 16 × 3 = 4 × BC
⇒ BC = 12 cm
Now shaded area = Area of bigger semicircle − Areas of smaller semicircles
= 1/2 π (8)2 - 1/2 π (2)2 - 1/2 π (6)2
= 32 π − 2 π −18 π = 12 π cm2
5-d; Let 5x and 4x be the length and breadth of the garden.
Then, 5x × 4x = 2000
⇒ x2 = 100 ⇒ x = 10
∴ Length = 50 m and breadth = 40 m
Let ‘d’ be the width of the road. Then,
(50 − 2d) (40 − 2d) = 2000 − 344 ⇒ d = 2 m
6-d; Let side of rectangle be 2x and x units. and side of square = y units
∴ 4y = 6x ⇒ x/y = 4/6 = 2/3
∴ 2x × x/y2 = 2x2/ y2 = 2 × 4/9 = 8 : 9
7-c; Area of square = a2 = 81 cm2 ⇒ a = 9 cm
Perimeter of square = 4 × a = 4 × 9 = 36 cm
Perimeter of semi circle = πr + 2r ⇒ 36 = r (π + 2)
⇒ 36 = r (36/7) ⇒ r = 36 × 7/36 = 7
Area of semicircle = 1/2 πr2
= 1/2 × 22/7 × 7 × 7 = 77 cm2
Hence, ungrazed area = 336 − 77 = 259 m2
11-b; Let the length of the rope be r. Horse can graze an area equal to area of the circle of radius r.
Then, πr2 = 154 ⇒ r = 7 m
12-b; Radius of the outer circle = 28 + 14 = 42 m
Area of the path = Area of outer circle − Area of inner circle
= π (42)2 − π (28)2 = 3080 m2
13-c; Let r be the radius of the circular plot.
Then, πr2 = 154 ⇒ r = 7 m
Circumference of the plot = 2 × 22/7 × 7 = 44 m
Cost of fencing the plot = Rs.44 × 2.75 = Rs.121
14-b; Side of the square = 
= 22 cm
Perimeter of the square = 4 × 22 = 88 cm
Perimeter of the circle = 2π × radius = 88 cm
⇒ Radius = 14 cm
Area of the circle = π × (14)2
= 22/7 × 14 × 14 = 616 cm2
15-c; Area of equilateral triangle = √3/4 a2
[where a is side of triangle]
∴ √3/4 a2 = 121√3 ⇒ a = 22 cm
Perimeter of triangle = 22 × 3 = 66 cm
Perimeter of circle = 2πr i.e. 2πr = 66
∴ Area of outer circle = πr2 = 22/7 × 21 × 21 = 1386 cm2
area of inner circle = πr12 = 22/7 × 14 × 14 = 616 cm2
Hence, area of ring = 1386 − 616 = 770 cm2
