Mensuration
Area Of Trapezium And Rhombus
1
● Area of trapezium = × ( sum of parallel sides ) × ( distance between them).
2
2× Area of trapezium
● Height of a trapezium =
(∑ of ∥sides)
2× Area of trapezium
● Sum of parallel sides =
( Height of trapezium)
1
● Area of rhombus = × d1 × d2 ; where d1 and d2 are the lengths of the diagonals of the
2
rhombus.
1
● Side of a rhombus =
2 √ d 21 × d 22
Part - 1
1. Calculate the area enclosed by the adjoining figure.
2. In the adjoining figure AD is parallel to BC. Find the area of the trapezium ABCD.
3. Two parallel sides of a trapezium are in the ratio 7 : 11 and the distance between them is
17 cm. If the area of the trapezium is 306 cm2 , find the lengths of its parallel sides.
4. The diagonal of a rhombus are 8 cm and 15 cm. Find its area.
5. The area of a rhombus is 360 cm2 and one of the diagonal is 18 cm. Find the other
diagonal.
6.Each side of a rhombus is 13 cm and one diagonal is 10 cm. Find the length of its other
diagonal and the area of the rhombus.
7. The area of a trapezium is 360 m2, the distance between two parallel sides is 20 m and one
of the parallel side is 25 m. Find the other parallel side.
8. Find the area of a rhombus whose side is 6.5 cm and altitude is 5 cm. If one of its diagonal
is 13 cm long, find the length of other diagonal.
9. Find the height of a trapezium, the sum of the lengths of whose bases is 60 cm and whose
area is 600 cm2.
10. The area of a rhombus is 144 cm2. If one of its diagonal is twice as long as the other, find
the lengths of the diagonals.
11. The area of a trapezium is 1512 cm2. If the ratio of the parallel sides is 9 : 5 and the
distance between them is 24 cm, find the length of the parallel sides
Part - 2
1. The area of a trapezium is 168 cm2 and its height is 8 cm. If one of the parallel sides is
longer than the other by 6 cm, find the length of each of the parallel sides.
2. The parallel sides of a trapezium are 25 cm and 13 cm; its non parallel sides are equal,
each being 10 cm. Find the area of the trapezium.
3. The area of a trapezium with equal non-parallel sides is 168 m2. If the lengths of the
parallel sides are 36 m and 20 m, find the length of the non-parallel sides.
4. The parallel sides of a trapezium are 78 cm and 52 cm, while its non-parallel sides are 30
cm and 28 cm. Find the area of the trapezium.
5.A field is in the form of a trapezium whose parallel sides are 45.2 m and 22.8 m long and
the distance between them is 12 m. Find the cost of leveling it at Rs 3 per square metre.
6. A trapezium with parallel sides of length as 7: 3 is cut from a rectangle 30 dm by 4 dm so
as to have an area of one-third the latter. Find the length of the parallel sides.
7. A parallelogram and a rhombus are equal in area. The diagonals of a rhombus are 48 m and
36 m respectively. If one of the sides of the parallelogram is 54 m, find its corresponding
altitude.
8. From the adjoining diagram, calculate
(i) the area of trapezium ACDE
(ii) the area of parallelogram ABDE
(iii) the area of ∆BCD
C
9. The area of a rhombus is equal to the area of a triangle whose base and the corresponding
altitude are 24.8 cm and 16.5 cm respectively. If one of the diagonals of the rhombus is 22
cm, find the length of the other diagonal.
10. The perimeter of a trapezium is 52 cm. If its non-parallel sides are 10 cm each and its
altitude is 8 cm, find the area of the trapezium.
Answers
Part - 1
1. 145 cm2 2. 264 cm2
3. 14 cm, 22 cm 4. 60 cm2
5. 40 cm 6. 24 cm, 120 cm2
7. 11 m 8. 32.5 cm2, 5 cm
9. 20 cm 10. 12 cm, 24 cm
11. 81 cm, 45cm
Part - 2
1. 24 cm 2. 152 cm2 3. 10 m 4. 1680 cm2 5. Rs 1224
6. 14 dm , 6 dm 7. 16 m 8. (i) 65 m2 (ii) 45.5 m2 (iii) 19.5 m2
9. 18.6 cm 10. 128 cm2
Volume and Surface Area of Cube,
Cuboid and Cylinder
Comparision of volume and capacity:-
volume capacity
1cm3 1 ml
1000 cm
3
1000 ml∨1l
1m3 1000 l∨1 Kl
Cuboid:-
For a cuboid of length l units, breadth b units and height h units,
Volume of the cuboid = l b h cu units
Lateral surface area of the cuboid =2 h(l+b) sq units
Total surface area of the cuboid=2(lb+bh+ hl) sq units
Diagonal of the cuboid=√ l 2+ b2 +h2units
Cube :-
For a cube of side s units,
Volume of the cube = s3 cu unit
Lateral surface area of the cube = 4 s2 sq unit
Total surface area of the cube= 6 s2 sq unit
Diagonal of the cube=√ 3 s units
Cylinder:-
For a cylinder of height h and base radius r units;
Volume of the cylinder = r 2 h cu units
Curved surface area of the cylinder =2 rh sq units
Total surface area of the cylinder = 2 r (r + h) sq units
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22
Part -I (take as 7 )
1. If the total surface area of a cuboid is 432 sq m and the ratio of length, breadth and
height is 6 : 4 :3, then find the height of the cuboid.
2. Three cubes metal with edges 6 cm , 8 cm and 10 cmrespectively are melted to form a
single cube . Find the lateral surface area of the new cube formed.
3. If the total surface area of a cube is 24 sq cm , then find its volume .
4. Find the surface area and lateral surface area a cuboid where length, breadth & height
are in ratio 7 : 4 :3 and with volume 2268 cu cm .
5. Three cubes of edges 3 cm , 4 cm and x cm are melted into a single cube. If the volume
of the new cube is 216 cu cm, find the value of x .
6. If the length of the diagonal of a cube is 8 √ 3 cm. Then find the surface area and
volume.
7. The perimeter of one face of a cube is 20 cm . find i) its total surface area ii) the
volume iii) the diagonal.
8. If the capacity of a water tank is 140 kilo litres and its length and breadth are 7 m and
5 m respectively, then find its depth.
9. Find the capacity of a container having volume 1000 cubic dm .
10. If 11cm3 of gold is drawn into a wire of 1 mm diameter. Find the length of the wire.
11. The internal measurements of a box are 20 cm ×16 cm ×24 cm. How many cubes of
edge 4 cm can be put into this box.
12. A cuboid has a total surface area of 149 sq m and lateral surface area of 135 sq m. Find
the area of this base.
13. The external diameter of an iron pipe is 25 cm and its length is 20 cm . If the thickness
of the pipe is 1 cm, find the total surface area of the pipe.
14. A chamber is 12m long, 8m wide and 5m high. How many people can be
accommodated in the chamber if each person requires a space of 5 m3?
15. A granary is in the shape of a cuboid measuring 8 m ×6 m× 3 m. If a bag of grain
occupies a volume of 0.75 m 3 , how many bags of grain can be stored in the granary
22
Part - 2 (take as 7 )
1. If the area of the three adjacent faces of a cuboid are x , y & , then find its volume in
terms of x , y ∧z .
2. If the height of a cylinder is tripled (keeping radius of base same), then find the ratio
of volume of new cylinder obtained to that of the original cylinder..
3. If two cylinders have the same volume and the ratio of their heights is 1 :3, find the
ratio of their radii.
4. If the volume of a cylinder is 4.62 cu metre and its radius is 70 cm. Find its height (in
metres).
5. If the radii of two right circular cylinders are in ratio 2 :3 and their heights are in the
ratio 5 :4 , find the ratio of their curved surfaces.
6. A soft drink is available in two packs: a tin can in the shape of a cuboid with internal
measurements
6 cm ×5 cm and a height of 12cm and a plastic cylinder with internal radius 4cm and
height 14cm . Which container has a greater capacity and by how much?
7. A cylindrical bucket, 7cm in radius is filled with water to some height. A solid cuboid
measuring 22cm × 14cm × 8 cm is immersed in the water. Find the height by which
water rises in the bucket?
8. A metallic cylindrical pipe has thickness 0.25 cm and outside radius 3.25 cm. If 1 cm3
of the metal has a mass 12g , find the mass of the 70cm long pipe.
9. Find the ratio of the volumes of one cubical and one cylindrical container with same
height and height equals to radius of base .
10. In a box of dimension 1 m×1.2 m ×1.5 m, how many cubical boxes can be placed of
side 3cm.
Answers
Part-I Part -I Part-II
1. 6m.
2. 576sq cm
8. 4m 1. √ xyz
9. 1 Kilo Litre or 2. 3:1
3. 8 cu cm
1000 L
4. 1098sq cm,594sq cm 3. √ 3 :1
10. 14m
5. 5cm 4. 3m
11. 120
6. 384 sq cm,512 cu cm 5. 5 :6
12. 7 sq m
7. 6. Cylinder,344 sq cm
13. 3168 sq cm
i. 150 sq cm 7. 16 cm
14. 96
ii. 125 cu cm 8. 4125gm or 4.125 kg
15. 192
iii. 5√ 3 cm 9. 1: or 7:22
10. 66000
