Given below are the Class 9 Maths Worksheets for Surface area and volume
(a) Concepts questions
(b) Calculation problems
(c) Long answer questions
Question 1
A cube and cuboids have the same volume. The dimensions of cuboids are in ratio 1 : 2 : 4. If the difference between the cost of painting the cuboids and cube whole surface area at rate of Rs 5 per m
^{2} is Rs. 80. Find their volumes.
Solution
Let each side of cube be x m.
Ratio of sides of cuboid is 1:2:4.
Let the sides be y ,2y and 4y m
Now as per question
Volume of cube= volume of cuboid
$x^3=y \times 2y \times 4y=8y^3$
or x=2y
Surface Area of Cube = $6x^2= 24y^2$
Surface Area of Cubiod=$2(lb+lh+bh) =2(2y^2 + 8y^2 + 4y^2)$
As per the question
$[2(2y^2 + 8y^2 + 4y^2) - 24y^2 ] \times 5 =80$
$20y^2 = 80$
y=2
Therefore Side of cube = 4 m
Volume of cube =$(4)^3=64 \ m^3$
Now the sides of cuboid=y,2y,4y=2,4,8m
Therefore Volume of cuboid =$2 \times 4 \times 8=64 \ m^3$
Question 2
A tent is in shape of right circular cylinder up to a height of 3m and a cone above it. The maximum height of tent above ground is 13.5m. Calculate the cost of painting the inner side of tent at rate of Rs. 3/sq m if radius of base is 14m.
Solution
Dimension of Cylinder
$H_1=3 \ m$
r=14 m
Dimension of Cone
$H_2= 13.5 - 3 = 10.5 \ m$
r= 14 cm
Curved Surface Area of Tent = Curved Surface Area of Cylinder + Curved Surface Area of Cone
$=2 \pi rh_1 + \pi r \sqrt {r^2 + H_2^2}$
=1034 sq m
So cost will be = 1034 * 3 =Rs 3102
Question 3
A school provides milk to students daily in cylindrical glasses of diameter 7cm. If glass is filled with milk up to a height of 12cm, find how many liters of milk is needed to serve 1600 students.
Solution
d= 7 cm, r= 3.5 cm
h=12 cm
Volume of Glass = $\pi r^2 h= 462 \ cm^3$
Volume of Glass for 1600 students= 1600 * 462=739200 cm^{3} = 739.2 litre
Question 4
Eight metallic spheres, each of radius 2 mm, are melted and cast into a single sphere. Calculate the radius of the new sphere
Solution
$\frac {4}{3} \pi R^3= 8 \frac {4}{3} \pi (2)^3$
R=4mm
Question 5
The diameter of a sphere is decreased by 30%. By what percent its surface area decreases?
Solution
Initial radius=r
Original surface area =$4 \pi r^2$
If the radius of the sphere is decreased by 30% then its radius becomes 7r/10
Then the surface area of the new sphere will be
$= 4 \pi (\frac {7r}{10})^2 =\frac {49}{100} \times 4 \pi r^2$
Now the surface area is decreased by
$=\frac {4 \pi r^2 - 4 \pi r^2 \times \frac {49}{100}}{4 \pi r^2}= \frac {51}{100}$
So the percentage of the change in surface area is 51%.
Question 6
In a cylinder radius is doubled and height is halved then find its curved surface area.
Question 7
A right- angled ABC with sides 3cm, 4cm & 5cm is revolved about fixed side of 4cm. find the volume of solid generated. Also total surface area of solid.
Question 8
If h,s and v are height, curved surface and volume of cone respectively, prove that
$3v h^3 + 9v^2 -s^2 h^2=0$
Solution
$s= \pi rl$
$v= \frac {1}{3} \pi r^2 h$
Taking LHS
$3v h^3 + 9v^2 -s^2 h^2$
=$3 (\frac {1}{3} \pi r^2 h) h^3 + 9 (\frac {1}{3} \pi r^2 h)^2 - (\pi rl)^2 h^2$
$=\pi ^2 r^2 h^4 + \pi ^2 r^4 h^2 - \pi ^2 r^2 h^2 l^2$
Now $l^2 = h^2 + r^2$
$=0$
Hence Proved
Question 9
An edge of a cube measure r cm. If the largest possible right circular cone is cut out of this cube, then find the volume of cone?
Solution
Diameter of Cone= r, Radius= r/2
Height =r
Volume of cone=$ \frac {1}{3} \pi r^2 h$
$= \frac {1}{3} \pi (r/2)^2 r$
$=\frac {1}{12} \pi r^3$
Question 11
How many planks of dimensions 5m × 25cm × 10cm can be stored in a pit which is 20 m long, 6m wide and 80 cm deep?
Question 12
The surface area of a sphere of radius 5cm is 5 times the area of the curved surface area of cone of radius 4cm. Find the height and volume of cone. ( π= 22/7)
Solution
Surface area of sphere= $4 \pi r^2= 400 \pi \ cm^2$
curved surface area of cone=$ \pi rl \ cm^2 =4 \pi l \ cm^2$
Now As per the question
$400 \pi= 5 \times 4 \pi l$
l=5 cm
Now we know that
$l^2 = h^2 + r^2$
or $h^2 = l^2 - r^2 =5^2 - 4^2 = 3^2$
h= 3 cm
Now Volume of Cone is given by
$V= \frac {1}{3} \pi r^2 h$
Substituting the values
V=50.29 cm^{3}
Question 13
A sweet shop has one spherical ladoo of radius 5cm with the same amount of material, how many ladoos of radius 2.5cm can be made?
Question 14
A sphere and right circular cylinder of same radius have equal volumes by what percentage does diameters of cylinder exceeds its height?
Solution
Let the radius of sphere= r= Radius of a right circular cylinder
As per to the question,
Volume of cylinder= volume of a sphere
$\pi r^2 h=\frac {4}{3} \pi r^3$
$h=\frac {4}{3}r$
Now Diameter of the cylinder = 2r
So, Inreased diameter from height of the cylinder =$2r-\frac {4}{3}r=\frac {2}{3} r$
Therefore percentage increase in diameter of the cylinder =$ \frac {(2/3)r \times 100}{(4/3)r=50$ %
Question 15
Curved surface area of an ice cream cone of slate height 12cm is 113.04m
^{2}. Find the base radius and height of cone ( π= 3.14)
Question 16
A semi circular sheet of metal of radius 14cm is to be bent to form an open conical cup. Find the capacity of cup?
Question 17
Two solid spheres made of same metal have weight 5920g and 740g respectively. Determine the radius of larger sphere, if the diameter of smaller one is 5cm.
Question 18
Each edge of cube is increased by 50%. Find the percentage increase in surface area of cube.
Solution
Let a be the edge
Original Surface Area= $6a^2$
If edge of cube is increased by 50%, then edge becomes = 1.5a
New Surface Area = $6 (1.5a)^2= 13.5a^2$
% increase = $\frac { 13.5a^2 - 6a^2}{6a^2} \times 100= 125$%
Question 19
A residential house society is built in 4000 sq m in area. It has an underground thank to collect the rain water the length, breadth and height of which are 50m, 40m and 4m respectively. If it rains at the rate of 2mm per minute for 5 hours, then calculate the depth of water in the tank. What value is depicted in problem?
Question 20
A rectangular tank is 225m × 162m at base with what speed should water flow into it through an aperture 60cm x 45cm so that level of water is raised by 20cm in 2.5 hours.
Question 21
The pillars of a temple are cylindrical shaped. If each pillar has a circular base of radius 20cm and height 7m, then find the quantity of concrete mixture used to build 20 such pillars. Also find cost of concrete mixture at the rate of Rs. 200 per m
^{3}. ( π=22/7)
Question 22
50 circular plates, each of radius 7cm and thickness 12cm are placed one above the other to form a solid right circular cylinder. Find the total surface area and volume of the cylinder
Question 23
Three solid sphere of iron whose diameters are 2cm, 12cm and 16cm respectively is melted into a single solid sphere. Find the radius of solid sphere.
Question 24
Coins of same size (say 5 rupee coin) are placed one above the other and cylindrical block is obtained. The volume of this block is 67.76cm
^{3}. Find no. of coins arranged in block, if thickness of each coin is 2mm and radius of each coin is 1.4cm.
Question 25
Metal spheres, each radius 2cm, are packed in a rectangular box of internal dimensions 16cm × 8cm × 8cm. When 16 spheres are packed, the box filled with preservative liquid. Find the volume of liquid.
Solution
Volume of Liquid = Volume of Box - 16 × Volume of one sphere
Volume of Box= LBH
Volume of Sphere = $\frac {4}{3} \pi r^3}$
Substituting the values
Volume of Liquid =488 cm^{3}
Question 26
A semi- circular sheet of metal of diameter 14cm is bent to form an open conical container. Find the capacity of container?
Solution
Radius of semi - circular piece= 14 cm
Circumference of Sheet= &pi R=44 cm
Now when semi- circular sheet is bent to form an open conical container.
Circumference of Base of Cone = Circumference of Sheet
$2 \pi r= 44$
r= 7 cm
Now Slant height of cone= Radius of the plate = 14 cm
Now height of cone can be derived from the formula
$l^2 = h^2 + r^2$
$h= 7 \sqrt 3 $ cm
Now Volume of Cone is given by
$V= \frac {1}{3} \pi r^2 h$
Substituting the values
V=622.3 cm^{3}
Question 27
A 4cm cube is cut into 1cm cubes. Calculate the total surface area of the small cubes. What is the ratio of surface area of small cubes to that of large cube?
Question 28
A cylindrical pipe opened at both the ends is made of iron sheet which is 2cm thick. If the outer diameter is 12cm and its length is 200cm. Find how many cm
^{3} of iron has been used in making the pipe.
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