Critical Thinking Questions: Mensuration: Cylinder, Cone and Sphere

Type I

Statements Based Questions

Q1: Consider a solid cylinder with a radius of 5 cm and a height of 10 cm. Determine which of the following statements are correct:
Statement 1: The curved (lateral) surface area of the cylinder is 2π × 5 × 10 cm².
Statement 2: The total surface area of the cylinder is 2π × 5 × (10 + 5) cm².
Statement 3: The volume of the cylinder is π × 5² × 10 cm³.
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) All 1, 2, and 3

Q2: A cylindrical can has a diameter of 14 cm and a height of 20 cm. The can is to be painted on the outside. Which of the following statements are correct regarding the paint needed for the exterior surface?
Statement 1: The area to be painted includes the curved surface area and the area of the circular base.
Statement 2: The curved surface area alone is 2π × 7 × 20 cm².
Statement 3: The total exterior surface area to be painted is 2π × 7 × (20 + 7) cm².
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) All 1, 2, and 3

Q3: Consider the formulas and calculations given in the examples for cylinders. Identify which of the following statements are correct:
Statement 1: The volume of a cylinder with a base radius of 7 cm and height of 5 cm is 770 cm³.
Statement 2: For a cylindrical roller to cover an area of 5500 m², it must roll exactly 200 times.
Statement 3: The total surface area needed for constructing a cylindrical tank with a radius of 35 cm and height of 1 m is 47.10 m².
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) All 1, 2, and 3

Q4: Which of the following statements are true regarding the cylindrical shapes described in the problems?
Statement 1: The lateral surface area of a cylinder with a radius of 5 cm and height of 5 cm is 94.2 cm².
Statement 2: The total surface area of a closed cylindrical petrol storage tank with a diameter of 42 m and height 4.5 m is 87.12 m².
Statement 3: The number of revolutions a cylindrical roller of radius 1.75 m and width 2 m must make to cover an area of 200 m².
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) All 1, 2, and 3

Q5: The slant height of a cone is 10 cm and its base radius is 6 cm. Calculate the radius of the sector of the circle used to make the cone. Which of the following is correct?
Statement 1: The radius of the base is 6 cm.
Statement 2: The total surface area of the cone is 80π cm².
Statement 3: The slant height is equal to the length of the arc of the sector used to make the cone.
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) All 1, 2, and 3

Q6: A tent is made from a canvas of 100 m² with 1 m² wastage. The tent is in the form of a right circular cone. Identify the correct statements:
Statement 1: The slant height of the tent is 5 m.
Statement 2: The height of the tent is 24 m.
Statement 3: The volume of the conical tent is 523.3 m³.
(a) Only 1
(b) Only 2
(c) Only 1 and 2
(d) All 1, 2, and 3

Q7: A hemisphere has a surface area of 3πr². What would be the surface area of a full sphere with the same radius?
Statement 1: The surface area of the full sphere would be 6πr².
Statement 2: The surface area of the full sphere would be 4πr².
Statement 3: The surface area of the full sphere would be twice the surface area of the hemisphere.
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) Only 2 and 3

Q8: Rashmi is constructing a bird-bath for her garden, which is in the shape of a cylinder with a hemispherical depression at the end. If the height of the cylinder is reduced by 15 cm and the diameter remains the same, how does the surface area of the tin sheet required for making the bird-bath change?
Statement 1: The curved surface area of the cylinder will decrease.
Statement 2: The curved surface area of the hemisphere will not change.
Statement 3: The total surface area required in the sheet will decrease.
(a) Only 1
(b) Only 2
(c) Only 1 and 3
(d) All 1, 2, and 3

Q9: A cylindrical rod with a diameter of 4 cm and height of 21 cm is melted and recast into spherical balls of radius 1 cm each. How many balls are formed?
Statement 1: The volume of the rod is π × 2² × 21 cm³.
Statement 2: The volume of each ball is 4/3 π × 1³ cm³.
Statement 3: 66 balls are formed.
(a) Only 1
(b) Only 2
(c) Only 1 and 2
(d) All 1, 2, and 3

Q10: If a sphere and a cylinder have the same volume, and the height of the cylinder is twice the radius of its base, which of the following is true about the radius of the sphere (rₛ) and the radius of the cylinder's base (r꜀)?
Statement 1: rₛ = r꜀
Statement 2: rₛ > r꜀
Statement 3: rₛ < r꜀
(a) Only 1
(b) Only 2
(c) Only 3
(d) Cannot be determined from the given information

Type II

Assertion & Reason Based Questions

Q11: Assertion (A): The lateral surface area of a cylinder can be unfolded into a rectangle.
Reason (R): The height of the cylinder becomes the breadth of the rectangle.
(a) A is true, R is false
(b) A is false, R is true
(c) Both A and R are true
(d) Both A and R are false

Q12: Assertion (A): The volume of a right circular cylinder with a radius of 7 cm and height of 24 cm is 3696 cm³.
Reason (R): The formula for the volume of a cylinder is V = πr²h.
(a) A is true, R is false
(b) A is false, R is true
(c) Both A and R are true
(d) Both A and R are false

Q13: Assertion (A): If a conical tent has a base radius of 7 m and slant height of 25 m, then the curved surface area of the tent is 550 m².
Reason (R): The formula for the curved surface area of a cone is πrl.
(a) A is true, R is false
(b) A is false, R is true
(c) Both A and R are true
(d) Both A and R are false

Q14: Assertion (A): If a hemisphere and a solid sphere have the same total surface area, then their volumes will also be equal.
Reason (R): Total surface area and volume are directly proportional for all geometric shapes such as spheres and hemispheres.
(a) A is true, R is false
(b) A is false, R is true
(c) Both A and R are true
(d) Both A and R are false

Q15: Assertion (A): The volume of a sphere increases from the integral of infinitely many infinitesimal spherical shells of varying thickness.
Reason (R): The volume of the sphere is obtained by integrating the surface area of shells.
(a) A is true, R is false
(b) A is false, R is true
(c) Both A and R are true
(d) Both A and R are false

Q16: Assertion (A): The surface area of a solid includes all the faces.
Reason (R): For any given solid, the surface area is a measure of the total area of the faces the solid occupies.
(a) A is true, R is false
(b) A is false, R is true
(c) Both A and R are true
(d) Both A and R are false

Q17: Total volume of a rocket model consisting of a cylinder, a cone, and a hemisphere is found by adding the individual volumes of the three shapes.
Assertion (A): The total volume of the rocket is V = πr²h + (1/3)πr²h + (2/3)πr³.
Reason (R): The volume of a cone is (1/3)πr²h and the volume of a hemisphere is (2/3)πr³.
(a) A is true, R is false
(b) A is false, R is true
(c) Both A and R are true
(d) Both A and R are false

Q18: A child reshapes a cone into a sphere while maintaining the volume. The height of the cone is 24 cm and the radius of its base is 6 cm.
Assertion (A): The radius of the sphere will be 6 cm.
Reason (R): The volume of a cone and a sphere are both calculated using the radius.
(a) A is true, R is false
(b) A is false, R is true
(c) Both A and R are true
(d) Both A and R are false

Q19: Assertion (A): If a hollow sphere of internal radius 6 cm is melted to form small cones of base radius 2 cm and height 8 cm, the number of cones formed will be less than 40.
Reason (R): The volume of the hollow sphere is less than the combined volume of 40 cones of given dimensions.
(a) A is true, R is false
(b) A is false, R is true
(c) Both A and R are true
(d) Both A and R are false

Q20: Assertion (A): When a solid sphere is completely submerged in water inside a cylindrical vessel, the rise in water level depends on the sphere's volume.
Reason (R): The displaced water volume is equal to the volume of the sphere due to the principle of flotation.
(a) A is true, R is false
(b) A is false, R is true
(c) Both A and R are true
(d) Both A and R are false