SUMMARY: This chapter focuses on calculating the surface areas and volumes of different 3D shapes. KEY TOPICS: surface area of a cuboid, surface area of a cylinder, volume of a cuboid, volume of a cylinder, surface area of a cone, volume of a cone, surface area of a sphere, 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 cuboid?
A2(lw + lh + wh)
Bl + w + h
Clw + lh + wh
D2(l + w + h)
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Correct answer: Option 1 — 2(lw + lh + wh)
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
If the radius of a sphere is doubled, how does its volume change?
AIt remains the same
BIt doubles
CIt increases by a factor of 4
DIt increases by a factor of 8
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Correct answer: Option 4 — It increases by a factor of 8
Q41 Mark
The surface area of a cone with a radius of 4 cm and a slant height of 5 cm is:
A20π cm²
B25π cm²
C30π cm²
D15π cm²
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Correct answer: Option 2 — 25π cm²
Q51 Mark
Convert 5000 cm³ to liters. How many liters is that?
A5 liters
B50 liters
C0.5 liters
D500 liters
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Correct answer: Option 1 — 5 liters
Short Answer Questions5 questions
Q63 Marks
Calculate the surface area of a cuboid with length 5 cm, breadth 3 cm, and height 4 cm.
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The surface area of a cuboid is given by the formula 2(lb + bh + hl). Substituting the values, Surface Area = 2(5*3 + 3*4 + 4*5) = 2(15 + 12 + 20) = 2(47) = 94 cm².
Q73 Marks
Find the volume of a cylinder with a radius of 3 cm and a height of 7 cm.
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The volume of a cylinder is calculated using the formula V = πr²h. Substituting the values, V = π(3)²(7) = π(9)(7) = 63π cm³, which is approximately 197.82 cm³.
Q83 Marks
What is the surface area of a cone with a radius of 4 cm and a slant height of 5 cm?
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The surface area of a cone is given by the formula SA = πr(l + r). Substituting the values, SA = π(4)(5 + 4) = π(4)(9) = 36π cm², which is approximately 113.10 cm².
Q93 Marks
A sphere has a radius of 6 cm. Calculate its volume.
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The volume of a sphere is given by the formula V = (4/3)πr³. Substituting the radius, V = (4/3)π(6)³ = (4/3)π(216) = 288π cm³, which is approximately 904.32 cm³.
Q103 Marks
Convert the volume of a cuboid measuring 2 m, 3 m, and 4 m into liters.
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First, calculate the volume in cubic meters: V = l × b × h = 2 × 3 × 4 = 24 m³. Since 1 m³ = 1000 liters, the volume in liters is 24 × 1000 = 24000 liters.
Long Answer Questions5 questions
Q116 Marks
A cuboid has a length of 10 cm, a width of 5 cm, and a height of 2 cm. Calculate the surface area and volume of the cuboid. Show all your workings.
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To calculate the surface area of the cuboid, we use the formula: Surface Area = 2(lw + lh + wh). Substituting the values, we get: Surface Area = 2(10*5 + 10*2 + 5*2) = 2(50 + 20 + 10) = 2(80) = 160 cm². For the volume, we use the formula: Volume = l × w × h. Thus, Volume = 10 × 5 × 2 = 100 cm³.
Q126 Marks
A cylinder has a radius of 3 cm and a height of 7 cm. Calculate the curved surface area and total surface area of the cylinder. Provide detailed calculations.
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The curved surface area (CSA) of a cylinder is given by the formula: CSA = 2πrh. Substituting the values, CSA = 2 × π × 3 × 7 = 42π cm². The total surface area (TSA) is given by TSA = 2πr(h + r). Thus, TSA = 2π × 3 × (7 + 3) = 60π cm². Therefore, CSA ≈ 131.88 cm² and TSA ≈ 188.5 cm² (using π ≈ 3.14).
Q136 Marks
A cone has a base radius of 4 cm and a height of 9 cm. Calculate the volume and surface area of the cone. Show all calculations clearly.
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The volume of a cone is calculated using the formula: Volume = (1/3)πr²h. Substituting the values, Volume = (1/3) × π × 4² × 9 = (1/3) × π × 16 × 9 = 48π cm³. The slant height (l) can be calculated using Pythagoras' theorem: l = √(r² + h²) = √(4² + 9²) = √(16 + 81) = √97 cm. The surface area is given by: Surface Area = πr(l + r) = π × 4 × (√97 + 4). Thus, Surface Area ≈ 4π(9.84) ≈ 123.84 cm².
Q146 Marks
A sphere has a radius of 5 cm. Calculate its surface area and volume. Provide detailed workings and explain the significance of these calculations in real-life applications.
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The surface area of a sphere is calculated using the formula: Surface Area = 4πr². Substituting the radius, Surface Area = 4π(5)² = 4π(25) = 100π cm². The volume of the sphere is given by: Volume = (4/3)πr³. Thus, Volume = (4/3)π(5)³ = (4/3)π(125) = (500/3)π cm³. In real-life applications, these calculations are significant in fields such as manufacturing, where the surface area affects material costs, and in packaging, where volume determines capacity.
Q156 Marks
A water tank is in the shape of a cylinder with a diameter of 1.2 m and a height of 2.5 m. Calculate the volume of water it can hold and the total surface area of the tank. Explain how this calculation is useful in practical scenarios.
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The radius of the cylinder is half of the diameter, so r = 1.2/2 = 0.6 m. The volume of the cylinder is calculated using the formula: Volume = πr²h = π(0.6)²(2.5) = π(0.36)(2.5) = 0.9π m³. The total surface area is given by: TSA = 2πr(h + r) = 2π(0.6)(2.5 + 0.6) = 2π(0.6)(3.1) = 3.72π m². These calculations are crucial for determining the amount of water the tank can store and for planning the construction and maintenance of water supply systems.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The surface area of a cuboid is calculated using the formula 2(lb + bh + hl).
Reason (R): This formula accounts for all six faces of the cuboid.
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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 cylinder can be found using the formula πr²h.
Reason (R): This formula is derived from the area of the base multiplied by the height.
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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): The surface area of a cone is given by the formula πr(r + l), where l is the slant height.
Reason (R): The formula includes the base area and the lateral surface area of the cone.
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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 sphere is calculated using the formula (4/3)πr³.
Reason (R): This formula is applicable only for cylindrical shapes, not spherical shapes.
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Correct answer: Option 3 —
A is true, but R is false.
Q201 Mark
Assertion (A): When converting units of volume from cubic centimeters to cubic meters, you multiply by 1000.
Reason (R): Cubic meters are larger than cubic centimeters, so you divide by 1000.
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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 cuboid is given by the formula 2(lw + lh + wh).
Statement 2: The volume of a cuboid is calculated using the formula V = l × w × h.
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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 the formula V = πr²h.
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Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: The surface area of a cone can be found using the formula πr(l + r) where l is the slant height.
Statement 2: The volume of a cone is calculated using the formula V = 1/3πr²h.
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Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: The surface area of a sphere is given by the formula 4πr².
Statement 2: The volume of a sphere is calculated using the formula V = 2/3πr³.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q251 Mark
Statement 1: When converting units, 1 cm³ is equal to 1000 m³.
Statement 2: Real-life applications of surface area and volume include calculating the amount of paint needed for a wall.
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Correct answer: Option 4 —
Both statements are false.
