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MENSURATION-IV
Theory:
A solid is a figure bounded by one or more surface. Hence a solid has length,
breadth and height. The plane surfaces that bind a solid are called its faces. The
fundamental difference between a plane figure and a solid figure is that the plane
figure lies in a plane and a solid figure lies in space.
There are two types of three-dimensional figures
(1) The solid figure in which any of the cross section is the same throughout.
E.g. Cube, Cuboid, Cylinder etc.
(2) The solid figure in which none of the cross-sections is same throughout.
E.g. Cone, Sphere, Pyramid etc.
CUBOID:
A cuboid is bounded by 6 rectangular faces. The opposite faces of a rectangular
solid are equal rectangles lying in parallel planes.
E
F
A
B
h
b G
H
C
D
The areas of three different faces be A1 , A2 and A3 then A1 = lb
A2 = bh
A3 = lh
Surface area = 2 ( A1 + A2 + A3 ) = 2 (lb + bh + lh)
Volume = Area of any face × corresponding height
V = lb × h = lbh
Diagonal (d) = l 2 + b 2 + h 2
Diagonal is the biggest possible dimension of a cuboid.
Also A1 × A2 × A3 = (lb ) (bh ) (lh ) = (lbh ) 2 = V 2
V=
A1 A2 A3
CUBE:
a
a
a
A Cube is bounded by six square faces i.e. if the length ,breadth and height
of a cuboid are all equal then it is called a cube.
If each side of the cube is of ‘a’ units,
then its surface area(S.A) =6a2 and Its Volume(V) = a3
Diagonal of cube will be d =

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PROBLEMS
1. Each edge of a cube is decreased by 20%. The percentage of decrease in the
surface area of the cube is
1) 44%
2) 36%
3) 20%
4) 60%
5) None of
these
ANSWER: 2
Edge of the cube be 5 then its surface area = 6 × 52 = 150
4
100 − 20
After reduction new edge of the cube =
× 5 = × 5/ = 4
5/
100
New surface area of the cube = 6 × 42 = 96
150 − 96
54
Surface area reduces by
× 100 =
× 100 = 36%
150
150
Shortcut method:
(i) If each edge of a cube increased by x % then the surface area increases by
x2
S = 2 x +
%
100
(ii) If each edge of a cube decreased by x % then the surface area decreases by
x2
S = 2 x −
%
100
20 2
In the above problem x = 20%, then S = 2 × 20 −
% = (40 – 4)% = 36%
100
2. A cuboid (3 cm × 4 cm × 5 cm) is cut into unit cubes. The ratio of the total
surface area of all the unit cubes to that of the cuboid is
1) 180 : 3
2) 180 : 9
3) 180 : 36
4) 180 : 47
5) None of
these
ANSWER: 4
The dimensions of a cuboid are 3 × 4 × 5
Its surface area (S.A) = 2(3×4 + 4×5 + 3×5) = 94 cm2
If the cuboid is cut into unit cubes, then the number of unit cubes so formed = 3
×4×5
= 60
But surface area of each unit cube = 6 × 12 = 6
Total surface area of unit cubes = 6 × 60 = 360
Required ratio = 360 : 94 = 180 : 47
3. If the diagonal of a cube is 10 3 cm, then its surface area will be
2) 550 cm2
3) 600 cm2
4) 650 cm2
1) 500 cm2
these
ANSWER: 3
Diagonal (d) of a cube = 10 3
d
10 3
Its side a =
= 10
=
3
3
Surface area (S.A) of cube = 6a2 = 6 × 102 = 600 cm2
5) None of

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4. If the volume of a cube is 216 cm 2 , then the surface area of the cube will be
2) 216 cm 2
3) 218 cm 2
4) 220 cm 2
5) None of
1) 214 cm 2
these
ANSWER:
Volume (V) of a cube = a3 = 216
a = 3 216 = 6
Its surface area = 6a2 = 6 × 62 = 216 cm2
5. If six cubes, each of 10 cm edge, are joined end to end, then the surface area of
the resulting solid will be
1) 3600 cm 2
2) 3000 cm 2
3) 2600 cm 2
4) 2400 cm 2
5) None of
these
ANSWER: 3
When six cubes are joined end to end, a cuboid will be formed whose length is
6×10 = 60
cm, breadth 10 cm and height 10 cm respectively i.e. l = 60, b = 10 & h = 10
Surface area of cuboid = 2 (60×10 + 10×10 + 60×10)
= 2(600 + 100 + 600) = 2600 sq cm
6. If three cubes of copper, each with an edge of 6 cm, 8 cm and 10 cm respectively
are melted to form a single cube, then the diagonal of the new cube will be
1) 18 cm
2) 19 cm
3) 19.5 cm
4) 20.8 cm
5) None of
these
ANSWER: 4
If three cubes are melted to form a single larger cube then the volume of larger
cube so formed will be equal to the sum of the volumes of the three cubes.
Volume of the larger cube = 63 + 83 + 103 = 216 + 512 + 1000 = 1728
Side of larger cube = 3 1728 = 12
Diagonal of larger cube = 12 3 = 12 × 1.732 = 20.8 cm
7. A swimming pool 9 m wide and 12 m long is 1 m deep on the shallow side and 4
m deep on the deeper side. Its volume is
1) 408 m3
2) 360 m3
3) 270 m3
4) 208 m3
5) None of
these
ANSWER: 3
The cross-section of the swimming pool is a trapezium whose parallel sides are 1
m and 4 m and having a perpendicular distance of 9 m.
1+ 4
Area of cross-section =
× 9 = 22.5 sq m
2
Volume of swimming pool = 22.5 × 12 = 270 cu.m
8. The length, breadth and height of a cuboid are in the ratio 1 : 2 : 3. The length,
breadth and height of the cuboid are increased by 100%, 200% and 200%
respectively. Then the increase in the volume of the cuboid is
1) 5 times
2) 6 times
3) 12 times
4) 17 times
5) None of
these
ANSWER: 4
Length, breadth and height of cuboid be x , 2 x and 3 x respectively, then its
volume = x × 2 x × 3 x = 6 x 3

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When length, breadth and height are increased by 100%, 200% and 300%
respectively,
100 + 100
New length =
× x = 2x
100
100 + 200
New breadth =
× 2x = 6x
100
100 + 300
New height =
× 3x = 9 x
100
New volume = 2 x × 6 x × 9 x = 108 x 3
108 x 3 − 6 x 3 102 x 3
= 17 times
Increase in volume =
=
6x3
6x3
9. A cube of lead with edges measuring 6 cm each is melted and formed into 27
equal cubes. What will be the length of the edges of the new cubes?
1) 3 cm
2) 4 cm
3) 2 cm
4) 1 cm
5) None of these
ANSWER: 3
The edge of each smaller cube be ‘a’.
Then total volume of 27 cubes = 27a3
But total volume of 27 cubes is equal to volume of cube of edge 6 cm
27a3 = 63 = 216
216
=8
a3 =
27
a=2
10. The edges of a cuboid are in the ratio 1 : 2 : 3 and its surface area is 88 cm2. The
volume of the cuboid is
1) 120 cm3
2) 64 cm3
3) 48 cm3
4) 24 cm3
5) None of
these
ANSWER: 3
The edges of cuboid are in the ratio of 1 : 2 : 3. So the edges can be assumed as x ,
2 x and
3x
Surface area (S.A) = 2( x × 2 x + 2 x × 3 x + x × 3 x ) = 2(11x 2 ) = 22 x 2
22 x 2 = 88
x=2
The dimensions of cuboid will be 2, 4 and 6. The volume of cuboid = 2×4×6 =
48 cm3.
11. The areas of three adjacent faces of a cuboid are a,b and c. If the volume of the
cuboid is V, then V2 is equal to
c
4) (a + b + c)
5) None of
1) abc
2) (ab + bc + ca) 3)
ab
these
ANSWER: 1
If the three adjacent dimensions are x , y and z, then x × y = a y × z = b x × z = c
x y = a yz = b x z = c
( x y) (yz) (z x ) = abc

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