SUMMARY: This chapter focuses on calculating the surface areas and volumes of different three-dimensional shapes. KEY TOPICS: surface area of a cuboid, surface area of a cylinder, surface area of a cone, surface area of a sphere, volume of a cuboid, volume of a cylinder, volume of a cone, volume of a sphere, conversion of units, real-life applications of surface area and volume calculations.
What is the formula for the surface area of a cube with side length 'a'?
A6a^2
B4a^2
C2a^2
D8a^2
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Correct answer: Option 1 — 6a^2
Q21 Mark
A cylinder has a radius of 3 cm and a height of 5 cm. What is its volume?
A45π cm³
B30π cm³
C15π cm³
D60π cm³
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Correct answer: Option 1 — 45π cm³
Q31 Mark
The radius of a sphere is doubled. How does its surface area change?
AIt remains the same
BIt doubles
CIt quadruples
DIt increases by 50%
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Correct answer: Option 3 — It quadruples
Q41 Mark
If the height of a cone is tripled and the radius remains the same, what happens to the volume?
AIt triples
BIt remains the same
CIt quadruples
DIt increases by 50%
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Correct answer: Option 1 — It triples
Q51 Mark
A cone has a base radius of 4 cm and a height of 9 cm. What is the total surface area of the cone?
A52π cm²
B36π cm²
C60π cm²
D48π cm²
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Correct answer: Option 1 — 52π cm²
Short Answer Questions5 questions
Q63 Marks
What is the formula for the surface area of a cube?
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The surface area of a cube is calculated using the formula 6a², where 'a' is the length of one side of the cube.
Q73 Marks
How do you find the volume of a cylinder?
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The volume of a cylinder can be found using the formula V = πr²h, where 'r' is the radius of the base and 'h' is the height of the cylinder.
Q83 Marks
Explain how to calculate the surface area of a cone.
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The surface area of a cone is calculated using the formula SA = πr(l + r), where 'r' is the radius of the base and 'l' is the slant height of the cone.
Q93 Marks
What is the relationship between the radius and height of a sphere and its volume?
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The volume of a sphere is given by the formula V = (4/3)πr³, which shows that the volume increases with the cube of the radius, meaning a small increase in radius leads to a significant increase in volume.
Q103 Marks
Derive the formula for the volume of a cone. What does it represent?
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The volume of a cone is derived as V = (1/3)πr²h, representing one-third of the volume of a cylinder with the same base and height, illustrating how the cone occupies less space compared to the cylinder.
Long Answer Questions5 questions
Q116 Marks
A cylindrical water tank has a radius of 3 meters and a height of 5 meters. Calculate the total surface area of the tank. Show your calculations and explain each step.
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To find the total surface area of the cylindrical tank, we use the formula: Total Surface Area = 2πr(h + r), where r is the radius and h is the height. Substituting the values, we have: Total Surface Area = 2 × π × 3 × (5 + 3) = 2 × π × 3 × 8 = 48π square meters. Therefore, the total surface area of the tank is approximately 150.8 square meters when π is taken as 3.14.
Q126 Marks
A cone has a base radius of 4 cm and a height of 9 cm. Calculate the volume of the cone and explain the steps involved in the calculation.
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The volume of a cone is calculated using the formula: Volume = (1/3)πr²h, where r is the radius and h is the height. Here, r = 4 cm and h = 9 cm. Substituting these values, we get Volume = (1/3) × π × (4)² × 9 = (1/3) × π × 16 × 9 = 48π cubic centimeters. Therefore, the volume of the cone is approximately 150.8 cubic centimeters when π is approximated as 3.14.
Q136 Marks
A sphere has a radius of 7 cm. Calculate its surface area and volume. Provide detailed calculations and reasoning for each step.
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The surface area of a sphere is given by the formula: Surface Area = 4πr², and the volume is given by Volume = (4/3)πr³. For a sphere with radius r = 7 cm, the surface area is Surface Area = 4 × π × (7)² = 4 × π × 49 = 196π square centimeters. The volume is Volume = (4/3) × π × (7)³ = (4/3) × π × 343 = (1372/3)π cubic centimeters. Thus, the surface area is approximately 615.75 square centimeters and the volume is approximately 1436.76 cubic centimeters when using π ≈ 3.14.
Q146 Marks
A rectangular prism has dimensions of length 10 cm, width 5 cm, and height 4 cm. Calculate the total surface area and volume of the prism, explaining each step of your calculations.
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To find the total surface area of a rectangular prism, we use the formula: Total Surface Area = 2(lw + lh + wh). Here, l = 10 cm, w = 5 cm, and h = 4 cm. Thus, Total Surface Area = 2(10 × 5 + 10 × 4 + 5 × 4) = 2(50 + 40 + 20) = 2 × 110 = 220 square centimeters. The volume is calculated using the formula: Volume = l × w × h = 10 × 5 × 4 = 200 cubic centimeters. Therefore, the total surface area is 220 square centimeters and the volume is 200 cubic centimeters.
Q156 Marks
A frustum of a cone has a height of 6 cm, a radius of the lower base of 5 cm, and a radius of the upper base of 3 cm. Calculate the volume of the frustum and explain your calculations.
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The volume of a frustum of a cone is calculated using the formula: Volume = (1/3)πh(R² + r² + Rr), where R is the radius of the lower base, r is the radius of the upper base, and h is the height. Here, R = 5 cm, r = 3 cm, and h = 6 cm. Substituting these values, we get Volume = (1/3)π × 6 × (5² + 3² + 5 × 3) = (1/3)π × 6 × (25 + 9 + 15) = (1/3)π × 6 × 49 = 98π cubic centimeters. Therefore, the volume of the frustum is approximately 307.76 cubic centimeters when π is approximated as 3.14.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The surface area of a cylinder is given by the formula 2πr(h + r).
Reason (R): This formula accounts for both the curved surface area and the area of the two circular bases.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The volume of a cone is equal to one-third the volume of a cylinder with the same base and height.
Reason (R): This is because the cone can be thought of as being formed by removing a portion of the cylinder.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): A sphere has a greater surface area than a cube with the same volume.
Reason (R): The sphere minimizes surface area for a given volume, making it more efficient than a cube.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q191 Mark
Assertion (A): The volume of a hemisphere is given by the formula (2/3)πr^3.
Reason (R): This formula is incorrect; the correct volume of a hemisphere is (2/3)πr^3.
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Correct answer: Option 3 —
A is true, but R is false.
Q201 Mark
Assertion (A): The surface area of a cube can be calculated by the formula 6a^2.
Reason (R): This formula is only applicable when the cube's side length is known.
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Correct answer: Option 4 —
A is false, but R is true.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The surface area of a cube is given by the formula 6a², where 'a' is the length of a side.
Statement 2: The volume of a cube is calculated using the formula V = a³.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: The surface area of a cylinder is calculated using the formula 2πr(h + r).
Statement 2: The volume of a cylinder is given by V = πr²h.
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Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: The volume of a cone is one-third the volume of a cylinder with the same base and height.
Statement 2: The surface area of a cone is given by πrl + πr², where 'l' is the slant height.
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Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: The total surface area of a sphere is given by the formula 4πr².
Statement 2: The volume of a sphere is calculated using the formula V = (4/3)πr³.
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Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: The surface area of a rectangular prism is calculated by the formula 2(lw + lh + wh).
Statement 2: The volume of a rectangular prism is given by V = l × w × h.
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Correct answer: Option 1 —
Both statements are true.
