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CLASS X : CHAPTER - 13 SURFACE AREAS AND VOLUMES

CLASS X : CHAPTER - 13 SURFACE AREAS AND VOLUMES

IMPORTANT FORMULAS & CONCEPTS

FORMULAS FOR SOLIDS
The following table lists the formulas for various geometric solids:

Name Lateral/Curved Surface Area Total Surface Area Volume
Cuboid \( 2h(l + b) \) \( 2(lb + bh + hl) \) \( lbh \)
Cube \( 4a^2 \) \( 6a^2 \) \( a^3 \)
Right Prism Perimeter of base \( \times \) height LSA + 2(area of base) Area of base \( \times \) height
Right Circular Cylinder \( 2\pi rh \) \( 2\pi r(r + h) \) \( \pi r^2h \)
Right Pyramid \( \frac{1}{2} \)(perimeter of base) \( \times \) slant height LSA + area of base \( \frac{1}{3} \) area of base \( \times \) height
Right Circular Cone \( \pi rl \) \( \pi r(l + r) \) \( \frac{1}{3}\pi r^2h \)
Sphere \( 4\pi r^2 \) \( 4\pi r^2 \) \( \frac{4}{3}\pi r^3 \)
Hemisphere \( 2\pi r^2 \) \( 3\pi r^2 \) \( \frac{2}{3}\pi r^3 \)

FRUSTUM OF A CONE
If a right circular cone is cut off by a plane parallel to its base, the portion of the cone between the cutting plane and the base is called a frustum of a cone. Let \( h \) be the height, \( l \) be the slant height, and \( r_1 \) and \( r_2 \) be the radii of the ends.

  • Slant Height (\( l \)): \( \sqrt{h^2 + (r_1 - r_2)^2} \)
  • Lateral Surface Area: \( \pi(r_1 + r_2)l \)
  • Total Surface Area: \( \pi\{(r_1 + r_2)l + r_1^2 + r_2^2\} \)
  • Volume: \( \frac{\pi}{3}(r_1^2 + r_1r_2 + r_2^2)h \)
  • Height of original cone (\( h_1 \)): \( \frac{hr_1}{r_1 - r_2} \)

MCQ WORKSHEET-I

  1. The surface area of a cuboid is
    1. \( 2(lb + bh + lh) \)
    2. \( 3(lb + bh + lh) \)
    3. \( 2(lb - bh - lh) \)
    4. \( 3(lb - bh - lh) \)
  2. The surface area of a cube if edge ‘a’ is
    1. \( 7a^2 \)
    2. \( 6a^2 \)
    3. \( 5a^3 \)
    4. \( 5a^2 \)
  3. The length, breadth and height of a room is 5m, 4m and 3m. The cost of white washing its four walls at the rate of Rs. 7.50 per \( m^2 \) is
    1. Rs. 110
    2. Rs. 109
    3. Rs. 220
    4. Rs. 105
  4. The perimeter of floor of rectangular hall is 250m. The cost of the white washing its four walls is Rs. 15000. The height of the room is
    1. 5m
    2. 4m
    3. 6m
    4. 8m
  5. Two cubes each of edge 12 cm are joined. The surface area of new cuboid is
    1. 140 \( cm^2 \)
    2. 1440 \( cm^2 \)
    3. 144 \( cm^2 \)
    4. 72 \( cm^2 \)
  6. It is required to make a closed cylindrical tank of height 1 m and base diameter 140cm from a metal sheet. How many square meters a sheet are required for the same?
    1. 6.45 \( m^2 \)
    2. 6.48 \( m^2 \)
    3. 7.48 \( m^2 \)
    4. 5.48 \( m^2 \)

MCQ WORKSHEET-II

  1. The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to move once over to level a playground. The area of the playground in \( m^2 \) is:
    1. 1584
    2. 1284
    3. 1384
    4. 1184
  2. The inner diameter of circular well is 3.5m. It is 10m deep. Its inner curved surface area in \( m^2 \) is:
    1. 120
    2. 110
    3. 130
    4. 140
  3. The curved surface area of a right circular cone of slant height 10 cm and base radius 7 cm is
    1. 120 \( cm^2 \)
    2. 220 \( cm^2 \)
    3. 240 \( cm^2 \)
    4. 140 \( cm^2 \)
  4. A conical tent is 10 m high and the radius of its base is 24 m. The slant height of tent is
    1. 26 m
    2. 28 m
    3. 25 m
    4. 27 m

MCQ WORKSHEET-III & IV

  1. A joker’s cap is in the form of cone of base radius 7 cm and height 24 cm. The area of sheet to make 10 such caps is
    1. 5500 \( cm^2 \)
    2. 6500 \( cm^2 \)
    3. 8500 \( cm^2 \)
    4. 3500 \( cm^2 \)
  2. The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. The ratio of surface area of the balloon in the two cases is:
    1. 4 : 1
    2. 1 : 4
    3. 3 : 1
    4. 1 : 3
  3. A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute?
    1. 4000 \( m^3 \)
    2. 40 \( m^3 \)
    3. 400 \( m^3 \)
    4. 40000 \( m^3 \)
  4. If the volume of a right circular cone of height 9 cm is \( 48\pi \text{ cm}^3 \), find the diameter of its base.
    1. 12 cm
    2. 10 cm
    3. 6 cm
    4. 8 cm
  5. The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?
    1. 1/64
    2. 1/32
    3. 1/16
    4. 1/48

MCQ WORKSHEET-V

  1. The volume of a cuboidal solid of length 8 m and breadth 5 m is 200 \( m^3 \). Find its height.
    1. 5 m
    2. 6 m
    3. 15 m
    4. 18 m
  2. Base radius of two cylinder are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. The ratio of their volumes is
    1. 27 : 20
    2. 25 : 24
    3. 20 : 27
    4. 15 : 20
  3. If base radius and height of a cylinder are increased by 100% then its volume increased by:
    1. 30%
    2. 40%
    3. 42%
    4. 33.1%
  4. The volume of a sphere is 524 \( cm^3 \). The diameter of sphere is
    1. 5cm
    2. 4cm
    3. 3cm
    4. 7cm

MCQ WORKSHEET-VII (FRUSTUM & COMBINATIONS)

  1. Small spheres, each of radius 2cm, are made by melting a solid iron ball of radius 6cm, then the total number of small spheres is
    1. 9
    2. 6
    3. 27
    4. 81
  2. A solid sphere of radius r cm is melted and recast into the shape of a solid cone of height r. Then the radius of the base of cone is
    1. 2r
    2. r
    3. 4r
    4. 3r
  3. The radii of the ends of a frustum of a cone 40 cm high are 38 cm and 8 cm. The slant height of the frustum of cone is
    1. 50 cm
    2. \( 10\sqrt{7} \) cm
    3. 60.96 cm
    4. \( 4\sqrt{2} \) cm
  4. The circular ends of a bucket are of radii 35 cm and 14 cm and the height of the bucket is 40 cm. Its volume is
    1. 60060 \( cm^3 \)
    2. 80080 \( cm^3 \)
    3. 70040 \( cm^3 \)
    4. 80160 \( cm^3 \)

PRACTICE QUESTIONS

  1. A cone of maximum size is carved out from a cube of edge 14 cm. Find the surface area of the cone and of the remaining solid left out after the cone carved out.
  2. A solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.
  3. A canal is 300 cm wide and 120 cm deep. The water in the canal is flowing with a speed of 20 km/h. How much area will it irrigate in 20 minutes if 8 cm of standing water is desired?
  4. Three cubes of a metal whose edges are in the ratio 3:4:5 are melted and converted into a single cube whose diagonal is \( 12\sqrt{3} \) cm. Find the edges of the three cubes.
  5. A bucket is in the form of a frustum of a cone and holds 28.490 litres of water. The radii of the top and bottom are 28 cm and 21 cm, respectively. Find the height of the bucket.
  6. From a solid cube of side 7 cm, a conical cavity of height 7 cm and radius 3 cm is hollowed out. Find the volume of the remaining solid.
  7. Two solid cones A and B are placed in a cylindrical tube. The ratio of their capacities is 2:1. Find the heights and capacities of cones. Also, find the volume of the remaining portion of the cylinder.
  8. Marbles of diameter 1.4 cm are dropped into a cylindrical beaker of diameter 7 cm containing some water. Find the number of marbles that should be dropped into the beaker so that the water level rises by 5.6 cm.
  9. A wall 24 m long, 0.4 m thick and 6 m high is constructed with the bricks each of dimensions 25 cm \( \times \) 16 cm \( \times \) 10 cm. If the mortar occupies 1/10th of the volume of the wall, then find the number of bricks used in constructing the wall.
  10. A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 4 cm and the diameter of the base is 8 cm. Determine the volume of the toy. If a cube circumscribes the toy, then find the difference of the volumes of cube and the toy.
  11. A building is in the form of a cylinder surmounted by a hemispherical dome. The base diameter of the dome is equal to 2/3 of the total height of the building. Find the height of the building, if it contains \( 67 \frac{1}{21} \text{ m}^3 \) of air.
  12. Water flows at the rate of 10m/minute through a cylindrical pipe 5 mm in diameter. How long would it take to fill a conical vessel whose diameter at the base is 40 cm and depth 24 cm?
  13. A solid iron cuboidal block of dimensions 4.4 m \( \times \) 2.6 m \( \times \) 1m is recast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.
  14. A milk container of height 16 cm is made of metal sheet in the form of a frustum of a cone with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk at the rate of Rs. 22 per litre which the container can hold.
  15. A rocket is in the form of a right circular cylinder closed at the lower end and surmounted by a cone with the same radius as that of the cylinder. The diameter and height of the cylinder are 6 cm and 12 cm, respectively. If the slant height of the conical portion is 5 cm, find the total surface area and volume of the rocket.
  16. A solid right circular cone of height 120 cm and radius 60 cm is placed in a right circular cylinder full of water of height 180 cm such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is equal to the radius of the cone.
  17. A pen stand made of wood is in the shape of a cuboid with four conical depressions and a cubical depression to hold the pens and pins, respectively. The dimension of the cuboid are 10 cm, 5 cm and 4 cm. The radius of each of the conical depressions is 0.5 cm and the depth is 2.1 cm. The edge of the cubical depression is 3 cm. Find the volume of the wood in the entire stand.
  18. A toy is in the form of a cone on a hemi-sphere of diameter 7 cm. The total height of the top is 14.5cm. Find the volume and total surface area of the toy.
  19. Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no colour on it. He wanted to colour it with his crayons. The top is shaped like a cone surmounted by a hemisphere. The entire top is 5 cm in height and the diameter of the top is 3.5 cm. Find the area he has to colour.
  20. A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm.
  21. A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of the milk which can completely fill the container, at the rate of Rs 20 per litre.
  22. An oil funnel made of tin sheet consists of a 10 cm long cylindrical portion attached to a frustum of a cone. If the total height is 22 cm, diameter of the cylindrical portion is 8 cm and the diameter of the top of the funnel is 18 cm, find the area of the tin sheet required to make the funnel.
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