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CBSE CLASS IX

Surface Area and Volumes

General Instructions:

(i) All questions are compulsory.

(ii) The question paper consists of 34 questions divided into four sections – A, B, C and D. Section A comprises of 10 questions of 1mark each, Section B comprises of 8 questions of 2 marks each, Section C comprises of 10 questions of 3 marks each and section D comprises of 6 questions of 4 marks each.

(iii) Question numbers 1 to 10in section A are multiple choice question where you are to select one correct option out of the given hour.

(iv) There is no overall choice. However, internal choice has been provided in 1questionof 2 marks, 3 questions of 3 marks each and 2 questions of 4 mark each. You have to attempt only one of the alternatives n all such question. Use of calculator is not permitted

1. Units of Measurement of Area and Volume

Length:

• 1 Centimetre (cm) = 10 milimetre(mm)
• 1 Decimetre(dm) = 10 Centimetre
• 1 Metre = 10dm = 100 cm = 1000mm
• 1 Decametre (dam) = 10m = 1000cm
• 1 Hectometre (hm) = 10dam = 100m.
• 1 kilomemetre (km) =1000m = 100dam = 10hm.
• 1 Myriametre =10Kilometre

Area:

• 1cm² =1 cm X 1cm= 10 mm X 10mm = 100 mm²
• 1dm² =1 dm X 1dm=10 cm X 10 cm =100 cm²
• 1m² = 1 m X 1 m = 10 dm = 10 dm = 100 dm²
• 1 dam² or 1are = 1 dam X 1dam = 10m X 10m = 100 m²
• 1hm² = 1 hectare = 1hm X 1hm = 100 m X 100 m = 10000 m² = 100 dm²
• 1km² = 1 km X 1 km = 10 hm X 10 hm = 100 hm² or 100 hectare

Volume:

• 1 cm³ = 1 ml = 1 cm X 1 cm X 1 cm = 10 mm X 10 mm X 10 mm = 1000 mm³
• 1 Litre = 1000 ml =1000 cm³.
• 1 m³ = 1 m X 1 m X 1 m = 100 cm X 100 cm X 100 cm = 10⁶ cm³ = 1000 litre = 1 kilolitre
• 1 dm³ = 1000 cm³
• 1m³ = 1000 dm³
• 1 km³ = 10⁹m³

2. Surface Area and Volume of Cuboids

Let there be cuboids of Length (l), Breadth (b), Height (h), Area (A) and Volume (V):-

1. Total surface area of the cuboids = 2( lb + bh + lh ).
2. Lateral surface area of the cuboids= 2( l + b ) h,i.e. Product of (Perimeter of the base) and (Height).
3. Diagonal of the cuboids= √l²+b²+h² i.e. square root of ( l²+b²+h² ).
4. Length of all 12 edges of the cuboids= 4 (l+b+h).
5. Volume of a cuboids (V) =lbh, i.e. Product of (Area of the base) and (Height)
6. l = V / bh; b = V / lh; h = V / lb or height = Volume divide by area of the base
3. Surface Area and Volume of a Cube

If the length (l) of each edge, Area (A) and Volume (V) of a Cube:-

1. Total surface area of the cube = 6l²
2. Lateral surface area of the cube = 4l²
3. Diagonal of the cube =√3 l i.e. cube roots of length.
4. Length of all 12 edges of the cube = 12 l.
5. Volume of a Cube (V) = l³ = (edge)³ .
6. Edge of a cube= Cube roots of cube i.e.
4. Surface Area and Volumeof a Right Circular Cylinder
• Radius: The radius (r) of the circular base is called the radius of the cylinder.

• Height: The length of the axis of the cylinder is called the height (h) of the cylinder.

• Lateral Surface: The curved surface joining the two base of a right circular cylinder is called Lateral Surface.

• Hollow Cylinder: A solid bounded by two coaxial cylinders of the same height and different radii is called a hollow cylinder.

Radius (r), height (h), Area (A), and Volume (V) ;

1. Lateral or Curved surface area= 2πrh = Product of Circumference of the circle and height
2. Each Surface area = π r²
3. Total surface area of Right Circular cylinder= (2πrh + 2πr²) = 2πr (h+r).
4. Each Base surface area = π (R²-r²) {R= radius of outer cylinder, r=radius of inner cylinder).
5. Lateral or Curved surface area of hollow cylinder= (External surface area) + (Internal surface area) = 2πRh + 2πrh = 2πh (R+r).
6. Total surface area of hollow cylinder=2 π Rh+2 π rh=2 π ( R²-r² ) = 2πh (R+r) + 2π(R+r) (R-r) = 2π (R+r) (h+R-r)
7. Volume of the cylinder =Measure of the space occupied by the cylinder=Product of the area of each circular sheet and height= π r²h.
8. Volume of a Hollow Cylinder = Exterior Volume –Interior Volume= π R²h- π r²h= π (R²-r²)h

5. Surface Area and Volume of a Right Circular Cone
   • Base: a right circular cone has a plane end, which is in circular shape. This is called the base (π r²) of the cone.

   • Height: The length of the line segment joining the vertex to the centre of base is called the height (h) of the cone.

   • Slant Height: The length of the segment joining the vertex to any point on the circular edge of the base is called the slant height (l) of the cone.

   • Radius: The radius of the base circle is called the radius (r) of the cone.

   • The curved surface area of a cone is also called the lateral surface area.

   A Hollow Right Circular Cone of radius (r), height (h) and slant height (l), Area (A), Volume (V) then:

   1. Length of the circular edge=2 π r
   2. Area of the plane = π r²
   3. Curved surface area of the cone= Area of the Sector
      = 1|2 X (arc length) x ( radius)
      =1|2 x 2πr x l = πrl.
      = Half and (product of circumference of base) and (slant height)
   4. Total surface area of the cone =Curved surface area + Area of the base. = π rl+ π r²= π r (l+r).
   5. Volumes of a Right Circular Cone = 3 (Volume of the cone of radius r and height) = π r²h=1/3 (π r²) X h= 1/3 X (Area of the base) X (height).

6. Surface Area and Volume of a Sphere/ Hemisphere
   • Sphere: A sphere can also be considered as a solid obtained on rotating a circle About its diameter.

   • Hemisphere: A plane through the centre of the sphere divides the sphere into two equal parts, each of which is called a hemisphere.

   • Spherical Shell: The difference of two solid concentric spheres is called a spherical shell.

   • A spherical shell has a finite thickness, which is the difference of the radii of the two solid spheres which determine it.

   Sphere is the radius r, Area (A) and Volume (V);

   1. Surface areas of a sphere: S = 4 π r².
   2. Curved surface area of a hemisphere: S = 2 π r².
   3. Total surface area of a hemisphere: S = 2 π r²+ π r²=3 π r²
   4. If ‘R’ and ‘r’ are outer and inner radii of a hemisphere shell, then Outer surface area = 4 π R²
   5. Volume of a sphere V = 4/3 X (π r³).
   6. Volume of a hemisphere V = 2/3 X (π r³).
   7. Volume of the spherical shell whose outer and inner radii and ‘R’ and ‘r’ respectively, V = 4/3 π(R³-r³)

   Section A

   1. The curved surface area of a right circular cylinder of height 14 cm is 88cm². Find the diameter of the base of the cylinder.
   2. Curved surface area of a right circular cylinder is 4.4m². If the radius of the base of the cylinder is0.7m, find its height.
   3. Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24m.Find
      (i) the curved surface area and
      (ii) the total surface area of a hemisphere of radius 21 cm.
   4. The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas.
   5. Find the radius of a sphere whose surface area is 154cm².
   6. A matchbox measures 4cm x 2.5cm x 1.5cm. What will be the volume of a packet containing 12 such boxes?
   7. A cubical water tank is 6m long, 5m wide and 4.5m deep. How many liters of water can it hold?
   8. The capacity of a cubical tank is 50000 liters of water. Find the breadth of the tank, if its length and depth are respectively 2.3m and 10m.
   9. The height and the slant height of a cone are 21cm and 28 cm respectively. Find the volume of the cone.
   10. If the volume of a right circular cone of height 9cm is 48π cm3, find the radius of the base. (Use π=3.14)
       10(a) A triangle ABC with sides 5cm, 12cm and 13cm cm is revolved about the side 12 cm. Find the volume of the solid so obtained.
   11. Find the volume of a sphere whose surface area is 154cm².
   12. How many liters of milk can a hemispherical bowl of diameter 10.5 cm hold?
   13. Find the amount of the water displaced by a solid spherical ball of diameter
       (i) 28cm
       (ii) 0.21m
   14. Find the lateral or curved surface area of a closed cylindrical petrol storage tank that is 4.2m in diameter and 4.5 m high. How much steel wad actually used, if 1/12 th of the steel actually used was wasted in making the tank.

   Section B

   1. A capsule of medicine is in the shape of a sphere of diameter 3.5mm. How much medicine (in mm³) is needed to fill this capsule?
   2. A hemispherical tank is made up of an iron sheet 1cm thick. If the inner radius is 1m, then find the volume of the iron used to make the tank.
   3. The diameter of a metallic ball is 4.2cm. What is the mass of the ball, if the density of the metal is 8.9 g per cm³?
   4. A conical pit of top diameter 3.5m is 12m deep. What is its capacity in kilolitres?
   5. The volume of a right circular cone is 9856 cm³. If the diameter of the base is 28cm, find
      (i) height of the cone
      (ii) Slant height of the cone.
      (iii) Curved surface area of the cone.
   6. A heap of the wheat is the form of a cone whose diameter is 10.5m and height is 3m. Find its volume. The heap is to be covered by congas to protect it from rain. Find the area of the canvas required.
   7. The circumference of the base of a cylindrical vessel is 132 cm and height is 25 cm. How many liters of water can it hold?
   8. A solid cube of side 12 cm is cut into eight cubes of equal volume. What will be the side of the new cube? Also, find the ratio between their surface areas.\
   9. A river 3m deep and 40m wide is flowing at the rate of 2km per hour. How much water will fall into the sea in a minute?
   10. A hemispherical bowl is made of steel, 0.25cm thick. The inner radius of the bowl is 5cm. Find the ratio of their surface areas.
   11. A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost of painting the curved surface of the pillar at the rate of Rs. 12.50per m².
   12. The floor of a rectangular hall has a perimeter 250m If the cost of painting the four walls at the rate of Rs 10per m² is Rs 15000, find the height of the hall.

   Section C

   1. Hameed has built a cubical water tank with lid for his house, with each outer edge 1.5 m long. He gets the outer surface of the tank excluding the base, covered with square tiles of side 25 cm. Find how much he would spend for the tiles, if the cost of the titles is Rs 360 per dozen.
   2. A plastics box 1.5 m long wide and 65 cm deep is to made. It is opened at the top. Ignoring the thickness of the plastics sheet, determine:
      (i) The area of the sheet required for making the box.
      (ii) The cost of the sheet for it, if a sheet measuring the box.
   3. A small indoor greenhouse (herbarium) is made entirely of glass panes (including base) held together with tape. It is 30 cm long, 25 cm, wide and 25 cm high.
      (i) What is the area of the glass?
      (ii) How much of tape is needed for all 12 edges?
   4. Savitri had to make a model of cylindrical kaleidoscope for her science project. She wanted to use chart paper to make the curved surface of the kaleidoscope. What would be the area of chart paper required by her, if she wanted to make a kaleidoscope of length 25cm with a 2.5 cm radius? (Use π=22/7)
   5. A metal pipe is 77 cm long. The inner diameter of a cross section is 4cm, the outer diameter being 4.4 cm. Find its
      (i) inner curved surface area.
      (ii) Outer curved surface area.
      (iii) Total surface area.
   6. In a hot water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system.
   7. The hollow sphere, in which the circus motorcyclist performs his stunts, has a diameter of 7m. Find the area available to the motorcyclist for riding.
   8. A hemispherical dome of a building needs to be painted. If the circumference of the base of the dome is 17.6m, find the cost of painting it, given the cost of painting is Rs 5 per 100cm².
   9. A village, having a population of 4000, requires 150liters of water per head per day. It has measuring 20m x 15 x 6m. For how many days will the water of this tank last?
   10. It cost Rs. 2200 to paint the inner curved surface of a cylinder of a cylinder vessel 10 deep. If the cost of painting is at the rate of Rs 20per m², find
       (i) inner curved surface area of the vessel.
       (ii) Radius of the base,
       (iii) Capacity of the vessel.

    

Submitted By Amit Kumar Jha
Email Id : [email protected]