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Surface Areas and Volumes · Class 10 Mathematics: 116 MCQs with answers

All 116 multiple-choice questions from this kit, free and with no sign-up.

Full kit · everything Flashcards · 44 Quick Revision · 34

Warm-up · 31 questions

Q1. The surface area of a combined solid is found by

Beginner
A adding the two total surface areas B adding the surfaces that remain visible C subtracting the smaller from the larger D multiplying the two areas
Show answer & explanation

adding the surfaces that remain visible

The joined faces are hidden.

Q2. When a cone is placed on a cylinder of the same radius, the surfaces that disappear are

Beginner
A the cone's base and the cylinder's top B the cone's curved surface C the cylinder's base D nothing disappears
Show answer & explanation

the cone's base and the cylinder's top

They are pressed together at the join.

Q3. The curved surface area of a cylinder is

Beginner
A $\pi r^2 h$ B $2\pi r(r+h)$ C $\pi rh$ D $2\pi rh$
Show answer & explanation

$2\pi rh$

The side, rolled out flat, is a rectangle.

Q4. The total surface area of a cylinder is

Beginner
A $2\pi rh$ B $\pi r^2 h$ C $4\pi r^2$ D $2\pi r(r+h)$
Show answer & explanation

$2\pi r(r+h)$

The curved part plus two circles.

Q5. The curved surface area of a cone is

Beginner
A $\pi r h$ B $\pi r l$ C $\frac13\pi r^2 h$ D $2\pi r l$
Show answer & explanation

$\pi r l$

With $l$ the slant height.

Q6. In $\pi r l$, the letter $l$ stands for

Beginner
A the slant height B the vertical height C the radius D the circumference
Show answer & explanation

the slant height

The distance down the sloping side.

Q7. The surface area of a sphere of radius $r$ is

Beginner
A $2\pi r^2$ B $3\pi r^2$ C $4\pi r^2$ D $\frac43\pi r^3$
Show answer & explanation

$4\pi r^2$

Four times the area of its great circle.

Q8. The CURVED surface area of a hemisphere is

Beginner
A $2\pi r^2$ B $3\pi r^2$ C $4\pi r^2$ D $\pi r^2$
Show answer & explanation

$2\pi r^2$

Half a sphere's surface.

Q9. The surface area of a cube of edge $a$ is

Beginner
A $a^3$ B $4a^2$ C $12a$ D $6a^2$
Show answer & explanation

$6a^2$

Six equal square faces.

Q10. The surface area of a cuboid is

Beginner
A $lbh$ B $6lb$ C $4(l+b+h)$ D $2(lb + bh + hl)$
Show answer & explanation

$2(lb + bh + hl)$

Three pairs of equal faces.

Q11. Surface area is measured in

Beginner
A cubic units B length units C degrees D square units
Show answer & explanation

square units

It covers a two-dimensional skin.

Q12. If a question gives a DIAMETER of 14 cm, the radius to use is

Beginner
A 14 cm B 7 cm C 28 cm D 3.5 cm
Show answer & explanation

7 cm

Halve it.

Q13. The total surface area of a cube of edge 5 cm is

Beginner
A 125 cm² B 100 cm² C 150 cm² D 25 cm²
Show answer & explanation

150 cm²

$6 \times 25$.

Q14. The FIRST step in any question of this topic is to

Beginner
A compute the volume B square the radius C find π D decide which surfaces remain visible
Show answer & explanation

decide which surfaces remain visible

Everything else follows from that list.

Q15. Assertion (A): The total surface area of a cylinder is always larger than its curved surface area. Reason (R): The total adds the areas of the two circular ends.

Beginner
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, and R is the correct explanation of A

Two positive areas are added, so the total must be larger.

Q16. State the rule for finding the surface area of a combined solid, and explain why adding the two total surface areas is wrong.

Beginner
Show answer & explanation

—

Q17. Write down the curved and total surface areas of a cylinder, a cone and a hemisphere.

Beginner
Show answer & explanation

—

Q18. The volume of a combined solid equals

Beginner
A the sum minus the join B the sum of the volumes of the parts C the larger part only D the difference of the parts
Show answer & explanation

the sum of the volumes of the parts

Nothing is lost at a join.

Q19. A cylinder of radius $r$ and height $h$ has volume

Beginner
A $2\pi rh$ B $\pi r^2 h$ C $\frac13\pi r^2 h$ D $\pi r h^2$
Show answer & explanation

$\pi r^2 h$

Base area times height.

Q20. A cone of radius $r$ and vertical height $h$ has volume

Beginner
A $\pi r^2 h$ B $\pi r l$ C $\frac13\pi r^2 h$ D $\frac13\pi r^2 l$
Show answer & explanation

$\frac13\pi r^2 h$

A third of its cylinder.

Q21. How many cones fill the cylinder of the same base and height?

Beginner
A three B two C four D one
Show answer & explanation

three

Which is exactly why the formula has a third.

Q22. A sphere of radius $r$ has volume

Beginner
A $4\pi r^2$ B $\frac23\pi r^3$ C $\frac43\pi r^2$ D $\frac43\pi r^3$
Show answer & explanation

$\frac43\pi r^3$

Four thirds, and cubed.

Q23. A hemisphere of radius $r$ has volume

Beginner
A $\frac43\pi r^3$ B $2\pi r^2$ C $\frac23\pi r^3$ D $\frac13\pi r^3$
Show answer & explanation

$\frac23\pi r^3$

Half a sphere.

Q24. A cube of edge $a$ has volume

Beginner
A $6a^2$ B $3a$ C $a^2$ D $a^3$
Show answer & explanation

$a^3$

Length times breadth times height, all equal.

Q25. A cuboid of sides $l$, $b$ and $h$ has volume

Beginner
A $lbh$ B $2(lb+bh+hl)$ C $l+b+h$ D $l^2bh$
Show answer & explanation

$lbh$

The three dimensions multiplied.

Q26. After a cavity is hollowed out, the material left has volume equal to

Beginner
A the whole solid plus the cavity B the cavity alone C unchanged D the whole solid minus the cavity
Show answer & explanation

the whole solid minus the cavity

Subtract what was removed.

Q27. How many cubic centimetres make one litre?

Beginner
A 100 B 10 C 1000 D 1,000,000
Show answer & explanation

1000

A cube of side 10 cm.

Q28. Volume is measured in

Beginner
A cubic units B square units C length units D litres only
Show answer & explanation

cubic units

Three dimensions.

Q29. A cube of edge 5 cm has volume

Beginner
A 150 cm³ B 125 cm³ C 25 cm³ D 75 cm³
Show answer & explanation

125 cm³

$5^3$.

Q30. A cuboid 5 by 4 by 3 cm has volume

Beginner
A 94 cm³ B 12 cm³ C 60 cm³ D 47 cm³
Show answer & explanation

60 cm³

$5 \times 4 \times 3$.

Q31. Write down the volume formulas for a cylinder, cone, sphere, hemisphere, cube and cuboid.

Beginner
Show answer & explanation

—

Building up · 38 questions

Q32. Joining two solids together makes the total surface area

Intermediate
A larger B unchanged C smaller than the two apart D exactly double
Show answer & explanation

smaller than the two apart

Two faces are lost.

Q33. The slant height of a cone is

Intermediate
A $\sqrt{r^2 - h^2}$ B $r + h$ C $\sqrt{r^2 + h^2}$ D $\sqrt{h^2 - r^2}$
Show answer & explanation

$\sqrt{r^2 + h^2}$

Pythagoras on the radius and the vertical height.

Q34. A cone of radius 7 cm and vertical height 24 cm has slant height

Intermediate
A 25 cm B 31 cm C 17 cm D $\sqrt{31}$ cm
Show answer & explanation

25 cm

$\sqrt{49+576}$.

Q35. A cone of radius 5 cm and vertical height 12 cm has slant height

Intermediate
A 17 cm B 13 cm C 7 cm D $\sqrt{119}$ cm
Show answer & explanation

13 cm

$\sqrt{25+144}$.

Q36. The TOTAL surface area of a hemisphere is

Intermediate
A $2\pi r^2$ B $4\pi r^2$ C $\pi r^2$ D $3\pi r^2$
Show answer & explanation

$3\pi r^2$

The dome plus its flat circular face.

Q37. A hemisphere stuck onto another solid contributes

Intermediate
A $3\pi r^2$ B $2\pi r^2$ C $4\pi r^2$ D $\pi r^2$
Show answer & explanation

$2\pi r^2$

Its flat face is hidden at the join.

Q38. In that tent problem the floor is not counted because

Intermediate
A it has no area B it is curved C it is hidden by the join D it is not made of canvas
Show answer & explanation

it is not made of canvas

The question asks for canvas, and the ground is not canvas.

Q39. Assertion (A): The surface area of a cone uses the slant height, not the vertical height. Reason (R): The curved surface, unrolled, is a sector whose radius is the distance from the apex to the rim.

Intermediate
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, and R is the correct explanation of A

That distance is exactly the slant height.

Q40. When a cavity is made in a flat face, that face

Intermediate
A disappears entirely B is unchanged C loses a circle of area $\pi r^2$ D doubles
Show answer & explanation

loses a circle of area $\pi r^2$

Reduced, not removed.

Q41. A word like 'open' or 'without a lid' in a question means

Intermediate
A the solid has no volume B one circular face is left out C the radius is doubled D nothing changes
Show answer & explanation

one circular face is left out

There is no surface where the lid would be.

Q42. The curved surface of a cylinder of radius 7 cm and height 10 cm, with $\pi = \frac{22}{7}$, is

Intermediate
A 440 cm² B 308 cm² C 770 cm² D 220 cm²
Show answer & explanation

440 cm²

$2 \times \frac{22}{7} \times 7 \times 10$.

Q43. The curved surface of a hemisphere of radius 7 cm is

Intermediate
A 154 cm² B 616 cm² C 308 cm² D 462 cm²
Show answer & explanation

308 cm²

$2 \times \frac{22}{7} \times 49$.

Q44. The surface of a sphere of radius 7 cm is

Intermediate
A 308 cm² B 154 cm² C 1232 cm² D 616 cm²
Show answer & explanation

616 cm²

$4 \times \frac{22}{7} \times 49$.

Q45. A cone of radius 7 cm and slant height 25 cm has curved surface

Intermediate
A 154 cm² B 704 cm² C 550 cm² D 275 cm²
Show answer & explanation

550 cm²

$\frac{22}{7} \times 7 \times 25$.

Q46. An awkward fraction appearing part-way through such a calculation usually means

Intermediate
A π was wrong B a diameter was not halved C the solid was impossible D the units were wrong
Show answer & explanation

a diameter was not halved

Questions choose radii so that $\frac{22}{7}$ cancels.

Q47. A cuboid measuring 5 by 4 by 3 cm has surface area

Intermediate
A 94 cm² B 60 cm² C 47 cm² D 120 cm²
Show answer & explanation

94 cm²

$2(20 + 12 + 15)$.

Q48. Explain what the slant height of a cone is, how it is found, and why it is used in the curved surface formula.

Intermediate
Show answer & explanation

—

Q49. When does a hemisphere contribute $2\pi r^2$ to a surface area, and when $3\pi r^2$?

Intermediate
Show answer & explanation

—

Q50. A cone of radius 5 cm has vertical height 12 cm. Find its slant height and its curved surface area in terms of $\pi$.

Intermediate
Show answer & explanation

—

Q51. List the three questions worth asking about a combined solid before writing any formula.

Intermediate
Show answer & explanation

—

Q52. Explain why examination questions in this chapter tend to use radii of 7, 14 or 3.5 cm.

Intermediate
Show answer & explanation

—

Q53. Volumes add at a join, whereas surfaces do not, because

Intermediate
A a join hides two faces but occupies no space B volume is bigger C surfaces are curved D volumes are approximate
Show answer & explanation

a join hides two faces but occupies no space

The join has no thickness.

Q54. A cone's volume formula uses

Intermediate
A the slant height B the diameter C the circumference D the vertical height
Show answer & explanation

the vertical height

The slant height never appears in a volume.

Q55. A cylinder of radius 7 cm and height 10 cm has volume, with $\pi = \frac{22}{7}$,

Intermediate
A 440 cm³ B 770 cm³ C 1540 cm³ D 220 cm³
Show answer & explanation

1540 cm³

$\frac{22}{7} \times 49 \times 10$.

Q56. A cone of radius 21 cm and vertical height 10 cm has volume

Intermediate
A 13860 cm³ B 1540 cm³ C 2310 cm³ D 4620 cm³
Show answer & explanation

4620 cm³

$\frac13 \times \frac{22}{7} \times 441 \times 10$.

Q57. Assertion (A): When two solids are joined, no volume is lost. Reason (R): Volume is measured in cubic units while surface area is measured in square units.

Intermediate
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, but R is not the correct explanation of A

Both are true, but the difference in units does not explain why nothing is lost when two solids are joined.

Q58. How many litres are in one cubic metre?

Intermediate
A 100 B 1000 C 10 D 1
Show answer & explanation

1000

A metre is 100 cm, so a cubic metre is $10^6$ cm³.

Q59. A tank holding 7700 cm³ has capacity

Intermediate
A 77 litres B 0.77 litres C 770 litres D 7.7 litres
Show answer & explanation

7.7 litres

Divide by 1000.

Q60. Dividing by 100 instead of 1000 when converting cm³ to litres makes the answer

Intermediate
A ten times too small B correct C ten times too large D a hundred times too large
Show answer & explanation

ten times too large

The divisor is ten times too small.

Q61. An answer of '2772 m²' for the air inside a tent is

Intermediate
A correct B wrong, because the number is wrong C wrong, because the unit should be m³ D correct if rounded
Show answer & explanation

wrong, because the unit should be m³

Volume takes cubic units.

Q62. A cone and a hemisphere of the same radius are joined base to base. The volume is

Intermediate
A the difference of the two B the sum of the two volumes C the larger of the two D half the sum
Show answer & explanation

the sum of the two volumes

Volumes simply add.

Q63. A cylinder of radius 3.5 cm and height 8 cm has volume, with $\pi = \frac{22}{7}$,

Intermediate
A 308 cm³ B 176 cm³ C 88 cm³ D 616 cm³
Show answer & explanation

308 cm³

$\frac{22}{7} \times 12.25 \times 8$.

Q64. The commonest arithmetic slip in this topic is

Intermediate
A using π B adding instead of multiplying C squaring the radius D dropping the $\frac13$, $\frac23$ or $\frac43$
Show answer & explanation

dropping the $\frac13$, $\frac23$ or $\frac43$

Each of the three curved solids carries a fraction.

Q65. Explain why the volumes of two joined solids simply add, while their surface areas do not.

Intermediate
Show answer & explanation

—

Q66. Explain why a cone's volume is one third of its cylinder's, and state the 1 : 2 : 3 ratio.

Intermediate
Show answer & explanation

—

Q67. Explain which height a cone's volume uses, and contrast it with the curved surface formula.

Intermediate
Show answer & explanation

—

Q68. Give the conversions between cubic centimetres, cubic metres and litres, and convert 7700 cm³.

Intermediate
Show answer & explanation

—

Q69. Explain why using a sphere's diameter in place of its radius makes the volume eight times too large.

Intermediate
Show answer & explanation

—

Topper level · 47 questions

Q70. Two hemispheres on the two ends of a cylinder contribute, together,

Advanced
A $2\pi r^2$ B $6\pi r^2$ C $3\pi r^2$ D $4\pi r^2$
Show answer & explanation

$4\pi r^2$

Exactly one sphere's surface.

Q71. A capsule is a cylinder of radius 3.5 mm and length 8 mm with hemispherical ends. Its surface area, with $\pi = \frac{22}{7}$, is

Advanced
A 330 mm² B 176 mm² C 154 mm² D 484 mm²
Show answer & explanation

330 mm²

$176 + 154$.

Q72. A tent has a cylinder of radius 7 m and height 10 m under a cone of vertical height 24 m. The canvas needed is

Advanced
A 550 m² B 990 m² C 440 m² D 1144 m²
Show answer & explanation

990 m²

$\pi r l + 2\pi rh$ with $l = 25$.

Q73. Assertion (A): Scooping a hemisphere out of a solid reduces its surface area. Reason (R): A hemispherical cavity removes a circle of area $\pi r^2$ from the face and exposes a bowl of area $2\pi r^2$.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is false but R is true

R is true and shows A is false: the net change is $+\pi r^2$, so the surface INCREASES.

Q74. Hollowing a hemisphere out of a face changes the surface area by

Advanced
A $-\pi r^2$ B $+\pi r^2$ C $+2\pi r^2$ D 0
Show answer & explanation

$+\pi r^2$

Lose a circle, gain a bowl.

Q75. A hemisphere of the same radius sits on top of a cylinder. The surfaces counted are

Advanced
A the dome and the whole cylinder B everything twice C the dome, the cylinder's side and its base D only the dome
Show answer & explanation

the dome, the cylinder's side and its base

The cylinder's top and the hemisphere's flat face are buried.

Q76. A cone and a hemisphere of equal radius are joined base to base. The surfaces counted are

Advanced
A both total surfaces B the cone's slope and the hemisphere's dome C the two flat circles D only the cone
Show answer & explanation

the cone's slope and the hemisphere's dome

The two circular faces are pressed together.

Q77. Which figure would you use for a solid hemisphere resting flat on a table, if its whole outside is to be painted?

Advanced
A $3\pi r^2$ B $2\pi r^2$ C $4\pi r^2$ D $\pi r^2$
Show answer & explanation

$3\pi r^2$

Dome and flat face both count if the underside is painted.

Q78. Which quantity is NOT needed for the curved surface of a cone?

Advanced
A the radius B the vertical height C the slant height D the value of π
Show answer & explanation

the vertical height

$\pi r l$ contains no $h$ — though $h$ may be needed to FIND $l$.

Q79. Why does a sphere have the least surface area for a given volume?

Advanced
A it is the largest shape B π is small C it is the most symmetric shape, with no corners or edges D it has no faces
Show answer & explanation

it is the most symmetric shape, with no corners or edges

Any deviation from sphericity adds surface.

Q80. The curved surface of a hemisphere compared with its own flat face is

Advanced
A exactly twice as large B the same C half as large D three times as large
Show answer & explanation

exactly twice as large

$2\pi r^2$ against $\pi r^2$.

Q81. Assertion (A): Two hemispheres joined to the ends of a cylinder contribute the same surface as one whole sphere. Reason (R): Each hemisphere shows its curved part and its flat circular face as well.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is true but R is false

The flat faces are joined to the cylinder and are not exposed, so only the curved parts count. The assertion stands, the reason is false.

Q82. A cone of radius 3 cm and vertical height 4 cm has curved surface, in terms of $\pi$,

Advanced
A $12\pi$ cm² B $9\pi$ cm² C $15\pi$ cm² D $20\pi$ cm²
Show answer & explanation

$15\pi$ cm²

$l = 5$, so $\pi \times 3 \times 5$.

Q83. A medicine capsule is a cylinder of radius 3.5 mm and length 8 mm with a hemisphere on each end. Find its surface area, taking $\pi = \frac{22}{7}$.

Advanced
Show answer & explanation

—

Q84. A tent consists of a cylinder of radius 7 m and height 10 m with a cone of vertical height 24 m on top. Find the area of canvas needed, taking $\pi = \frac{22}{7}$.

Advanced
Show answer & explanation

—

Q85. Explain what happens to a solid's surface area when a hemispherical cavity is hollowed out of one of its flat faces.

Advanced
Show answer & explanation

—

Q86. (a) State the rule for the surface area of a combined solid and explain what happens at the join. (b) A toy is a cone of radius 7 cm and vertical height 24 cm standing on a hemisphere of the same radius. Find its total surface area, with $\pi = \frac{22}{7}$. (c) Compare your answer with the sum of the two separate total surface areas, and explain the difference.

Advanced
Show answer & explanation

—

Q87. (a) Explain the difference between the slant height and the vertical height of a cone, and where each is used. (b) A tent is a cylinder of radius 7 m and height 10 m with a cone of vertical height 24 m on top. Find the canvas required, with $\pi = \frac{22}{7}$. (c) If the canvas costs ₹70 per square metre, find the cost.

Advanced
Show answer & explanation

—

Q88. (a) Explain carefully what happens to a solid's surface area when a hemispherical cavity is hollowed out of a flat face. (b) A cubical block of edge 7 cm has a hemisphere of the largest possible diameter scooped out of one face. Find the surface area of the remaining solid, with $\pi = \frac{22}{7}$. (c) State whether the surface area rose or fell, and by how much.

Advanced
Show answer & explanation

—

Q89. (a) Write down the surface-area formulas for all six solids used in this chapter. (b) A solid is a cylinder of radius 3.5 cm and height 10 cm with a hemisphere of the same radius on each end. Find its surface area, with $\pi = \frac{22}{7}$. (c) Explain why the two flat circular faces do not appear anywhere in your calculation.

Advanced
Show answer & explanation

—

Q90. Which formula correctly pairs with the curved surface $\pi r l$?

Advanced
A the volume $\frac13\pi r^2 l$ B the volume $\pi r^2 l$ C the volume $\frac13\pi r^2 h$ D there is no pairing
Show answer & explanation

the volume $\frac13\pi r^2 h$

Slant for the side, vertical height for the inside.

Q91. On the same base and of the same height, cone : hemisphere : cylinder is

Advanced
A $1 : 1 : 1$ B $3 : 2 : 1$ C $1 : 2 : 3$ D $1 : 3 : 2$
Show answer & explanation

$1 : 2 : 3$

The fractions are $\frac13$, $\frac23$ and 1.

Q92. A tent of cylinder r = 7 m, h = 10 m with a cone of height 24 m on top holds

Advanced
A 1540 m³ B 1232 m³ C 2772 m³ D 990 m³
Show answer & explanation

2772 m³

$1540 + 1232$.

Q93. In that tent calculation, the number 25 (the slant height)

Advanced
A is used for the cone's volume B replaces 24 C is not used at all D is added to 24
Show answer & explanation

is not used at all

The slant height belongs to the surface area only.

Q94. A sphere of radius 7 cm has volume, with $\pi = \frac{22}{7}$,

Advanced
A $\frac{4312}{3}$ cm³ B 616 cm³ C $\frac{2156}{3}$ cm³ D 1372 cm³
Show answer & explanation

$\frac{4312}{3}$ cm³

$\frac43 \times \frac{22}{7} \times 343$.

Q95. A hemisphere of radius 3 cm has volume, in terms of $\pi$,

Advanced
A $36\pi$ cm³ B $9\pi$ cm³ C $12\pi$ cm³ D $18\pi$ cm³
Show answer & explanation

$18\pi$ cm³

$\frac23 \times \pi \times 27$.

Q96. A cavity has what effect on a solid's volume and surface area?

Advanced
A lowers the volume, raises the surface area B lowers both C raises both D raises the volume, lowers the surface area
Show answer & explanation

lowers the volume, raises the surface area

Material is lost but new inside surface is exposed.

Q97. Assertion (A): The volume of a cone is $\frac13\pi r^2 l$, where $l$ is the slant height. Reason (R): The slant height is longer than the vertical height.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

A is false but R is true

R is true, but the volume formula uses the VERTICAL height, so A is false.

Q98. A cubical block of edge 7 cm has a hemisphere of radius 3.5 cm scooped out. The wood remaining is about

Advanced
A 343 cm³ B 253.17 cm³ C 89.83 cm³ D 432.83 cm³
Show answer & explanation

253.17 cm³

$343 - \frac{539}{6}$.

Q99. A pipe has outer radius $R$, inner radius $r$ and length $h$. The MATERIAL in it occupies

Advanced
A $\pi h R^2$ B $\pi h(R^2 - r^2)$ C $\pi h r^2$ D $\pi h(R - r)^2$
Show answer & explanation

$\pi h(R^2 - r^2)$

Outer cylinder minus inner cylinder.

Q100. For that same pipe, the CAPACITY is

Advanced
A $\pi R^2 h$ B $\pi r^2 h$ C $\pi h(R^2 - r^2)$ D $2\pi rh$
Show answer & explanation

$\pi r^2 h$

What it can hold is the inner volume.

Q101. Using a sphere's DIAMETER in place of its radius makes the volume

Advanced
A twice too large B eight times too large C half as large D correct
Show answer & explanation

eight times too large

The radius is cubed, and $2^3 = 8$.

Q102. Two hemispheres joined to the ends of a cylinder contribute a volume of

Advanced
A $\frac43\pi r^3$ B $\frac23\pi r^3$ C $\frac83\pi r^3$ D $2\pi r^3$
Show answer & explanation

$\frac43\pi r^3$

Exactly one sphere.

Q103. A sphere fills what fraction of the smallest cylinder containing it?

Advanced
A half B two thirds C three quarters D one third
Show answer & explanation

two thirds

$\frac43\pi r^3$ against $2\pi r^3$ — a result Archimedes prized.

Q104. A cone's volume and its curved surface share which quantity?

Advanced
A the vertical height B the slant height C neither shares anything D the radius
Show answer & explanation

the radius

$r$ appears in both; $h$ only in the volume and $l$ only in the surface.

Q105. A cone of radius 3 cm and vertical height 7 cm has volume, with $\pi = \frac{22}{7}$,

Advanced
A 66 cm³ B 198 cm³ C 22 cm³ D 132 cm³
Show answer & explanation

66 cm³

$\frac13 \times \frac{22}{7} \times 9 \times 7$.

Q106. Assertion (A): Hollowing a cavity out of a solid lowers its volume and raises its surface area. Reason (R): Material is removed, but the inside of the cavity becomes an exposed surface.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
Show answer & explanation

Both A and R are true, and R is the correct explanation of A

The two effects happen at once, and in opposite directions.

Q107. Assertion (A): The capacity of a container is the same as the volume of material it is made from. Reason (R): A container with thick walls has an outer radius larger than its inner radius.

Advanced
A Both A and R are true, and R is the correct explanation of A B Both A and R are true, but R is not the correct explanation of A C A is true but R is false D A is false but R is true
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A is false but R is true

R is true and shows A false: capacity is the INNER volume, while the material is the difference of the two.

Q108. A hemisphere of radius 3.5 cm has volume, with $\pi = \frac{22}{7}$, is

Advanced
A $\frac{539}{6}$ cm³ B $\frac{539}{3}$ cm³ C 77 cm³ D $\frac{343}{3}$ cm³
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$\frac{539}{6}$ cm³

$\frac23 \times \frac{22}{7} \times 42.875$.

Q109. A tent is a cylinder of radius 7 m and height 10 m with a cone of vertical height 24 m on top. Find the air space inside, with $\pi = \frac{22}{7}$.

Advanced
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Q110. A cubical block of edge 7 cm has a hemisphere of radius 3.5 cm scooped out of one face. Find the volume of wood remaining, with $\pi = \frac{22}{7}$.

Advanced
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Q111. State what a cavity does to a solid's volume and to its surface area, and explain why the two differ.

Advanced
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Q112. A pipe has outer radius $R$, inner radius $r$ and length $h$. Distinguish its capacity from the volume of material in it.

Advanced
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Q113. (a) Explain why volumes add at a join while surface areas do not. (b) A medicine capsule is a cylinder of radius 3.5 mm and length 8 mm with a hemisphere on each end. Find its volume, with $\pi = \frac{22}{7}$. (c) Explain why the two flat circles, which were excluded from the surface-area calculation, make no difference here.

Advanced
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Q114. (a) State the volume formulas for the six solids of this chapter and point out where each fraction comes from. (b) A toy is a cone of radius 7 cm and vertical height 24 cm standing on a hemisphere of the same radius. Find its volume, with $\pi = \frac{22}{7}$. (c) State which of the numbers 24 and 25 you used and why.

Advanced
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Q115. (a) Explain how the volume of a solid with a cavity is found, and contrast the effect of a cavity on volume with its effect on surface area. (b) A cubical block of edge 7 cm has a hemisphere of the largest possible diameter scooped out of one face. Find the volume of wood remaining, with $\pi = \frac{22}{7}$. (c) Express the wood remaining as a percentage of the original block, to one decimal place.

Advanced
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Q116. (a) Give the conversions between cubic centimetres, cubic metres and litres, and explain the difference between capacity and the volume of material. (b) A cylindrical vessel of internal radius 7 cm and internal depth 20 cm is filled to the brim. Find its capacity in litres, with $\pi = \frac{22}{7}$. (c) Explain what would change if the question had given the OUTER radius as 7 cm and stated that the walls were 0.5 cm thick.

Advanced
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