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CBSE · Class 10 · Mathematics

Surface Areas and Volumes 12.3: Volume of a Combination of Solids

5 concepts · 40 practice questions with answers · Free to read

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Volume formulas and their relations

From daily life

Fill a cone with sand and pour it into a cylinder of the same base and height: it takes three cones to fill the cylinder.

Simple definition

For radius r and height h: cylinder πr²h, cone ⅓πr²h, sphere 4/3 πr³, hemisphere 2/3 πr³. When the height equals the radius, cone : hemisphere : cylinder = 1 : 2 : 3.

Formula: V = πr²h; ⅓πr²h; 4/3 πr³; 2/3 πr³

cone1hemisphere2cylinder3same r, height = r⅓πr³ : ⅔πr³ : πr³

In the figure: With r = h: cone, hemisphere and cylinder are in the ratio 1 : 2 : 3.

Rule 1: Single solids — Put r and h in and keep the π with 22/7 when r is a multiple of 7.

  • Easy: Sphere r = 21: 4/3 × 22/7 × 9261 = 38808 cm³.
  • Medium: Cone r = 7, h = 24: 1/3 × 154 × 24 = 1232 cm³.
  • Exam-type: Cylinder r = 7, h = 10: 154 × 10 = 1540 cm³.

Rule 2: Use the relations — One volume gives the others without recalculating.

  • Easy: Cylinder 216π (r = h = 6): cone 72π, hemisphere 144π.
  • Medium: Hemisphere is half of the sphere: 38808 ÷ 2 = 19404.
  • Exam-type: Three cones fill one cylinder of the same r and h.
Remember

The volume of a combination is the plain sum (or difference) of the volumes of its parts.

Common mistake

Using πr²h for a cone; it needs the factor ⅓.

Memory clue

"Cone a third, hemisphere two-thirds."

Practice questions

Easy
  1. 1With radius r = 10 cm (and height h = 13 cm where needed), which formula gives the volume of a cone?
  2. 2With radius r = 13 cm (and height h = 11 cm where needed), which formula gives the volume of a sphere?
  3. 3With radius r = 9 cm (and height h = 19 cm where needed), which formula gives the volume of a cone?
Medium
  1. 4With radius r = 17 cm (and height h = 7 cm where needed), which formula gives the volume of a hemisphere?
  2. 5A hemisphere and a cylinder each have base radius 4 cm and height 4 cm. Find the ratio of the volume of the hemisphere to the volume of the cylinder.
  3. 6A cone, a hemisphere and a cylinder all have radius r, and the cone and the cylinder both have height r. The cylinder's volume is 57 cm³. Find the volume of the cone (in cm³).
Challenging
  1. 7The radius of a hemisphere is increased from 5 cm to 25 cm. By what factor does its volume increase?
  2. 8A cone, a hemisphere and a cylinder all have radius r, and the cone and the cylinder both have height r. The cylinder's volume is 87 cm³. Find the volume of the cone (in cm³).
Show answers (try them yourself first)
  1. 11/3 πr²h
  2. 24/3 πr³
  3. 31/3 πr²h
  4. 42/3 πr³
  5. 52 : 3
  6. 619
  7. 7125
  8. 829

Cylinder with hemispheres (capsule, tank, vessel)

From daily life

How much medicine does a capsule hold? Add the middle cylinder and the two ends, which together make one sphere.

Simple definition

The volume of a cylinder with a hemisphere at each end is πr²h + 4/3 πr³, because two hemispheres make one sphere. With one hemisphere only, add 2/3 πr³. The capacity of a vessel is its inner volume.

Formula: V = πr²h + 4/3 πr³

cylinder h = 10r = 3πr²h + 4/3 πr³= 90π + 36π= 126π≈ 396

In the figure: Capsule r = 3, cylinder h = 10: 90π + 36π = 126π ≈ 396 cm³.

Live figure

Cylinder with hemispheres (capsule, tank, vessel)

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Rule 1: Capsule — Cylinder part plus one sphere.

  • Easy: r = 3, h = 10: cylinder π × 9 × 10 = 90π.
  • Medium: Two hemispheres = sphere 4/3 × π × 27 = 36π.
  • Exam-type: Total 126π = 126 × 22/7 = 396 cm³.

Rule 2: Vessel with one hemisphere — Cylinder plus one hemisphere.

  • Easy: r = 3, h = 10: cylinder 90π, hemisphere 2/3 × 27π = 18π.
  • Medium: Capacity 108π = 108 × 22/7 ≈ 339.4 cm³.
  • Exam-type: 1000 cm³ make 1 litre, so this is about 0.34 litre.
Remember

Write the cylinder part and the sphere part on separate lines before adding.

Common mistake

Using the total length as the cylinder height; subtract 2r first.

Memory clue

"Pipe in the middle, ball at the ends."

Practice questions

Easy
  1. 1A capsule is a cylinder of radius 11 cm and height 12 cm with a hemisphere of the same radius at each end. Find its volume (in cm³, in terms of π).
  2. 2A capsule is a cylinder of radius 8 cm and height 17 cm with a hemisphere of the same radius at each end. Find its volume (in cm³, in terms of π).
  3. 3Find the volume of a hemisphere of radius 16 cm (in cm³, in terms of π).
Medium
  1. 4A capsule is a cylinder of radius 9 cm and height 13 cm with a hemisphere of the same radius at each end. Find its volume (in cm³, in terms of π).
  2. 5Find the volume of a hemisphere of radius 17 cm (in cm³, in terms of π).
  3. 6A medicine capsule is a cylinder with a hemisphere at each end. Its total length is 14 mm and its diameter is 4 mm. Find its volume (in mm³, in terms of π).
Challenging
  1. 7A medicine capsule is a cylinder with a hemisphere at each end. Its total length is 19 mm and its diameter is 14 mm. Find its volume (in mm³, in terms of π).
  2. 8A capsule with hemispherical ends of radius 4 cm has volume 1552π/3 cm³. Find the height of its cylindrical part (in cm).
Show answers (try them yourself first)
  1. 19680π/3
  2. 25312π/3
  3. 38192π/3
  4. 42025π
  5. 59826π/3
  6. 6152π/3
  7. 72107π/3
  8. 827

Cone with hemisphere or cylinder (toy, ice-cream, tent)

From daily life

An ice-cream cone with a scoop on top is a cone plus a hemisphere; the air inside a tent is a cone plus a cylinder.

Simple definition

For a cone on a hemisphere: V = ⅓πr²h + ⅔πr³. For a cone on a cylinder: V = πr²H + ⅓πr²h, where H is the height of the cylinder and h that of the cone. Here h is the straight height; the slant height is not used for volume.

Formula: V = ⅓πr²h + ⅔πr³

cone h = 8, r = 6hemisphere r = 6⅓πr²h + ⅔πr³= 96π + 144π= 240π≈ 753.6

In the figure: Ice-cream r = 6, cone h = 8: 96π + 144π = 240π ≈ 753.6 cm³ (π = 3.14).

Live figure

Cone with hemisphere or cylinder (toy, ice-cream, tent)

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Rule 1: Cone with a hemisphere — The hemisphere has the same r as the cone’s base.

  • Easy: r = 6, h = 8: cone 1/3 × π × 36 × 8 = 96π.
  • Medium: Hemisphere 2/3 × π × 216 = 144π.
  • Exam-type: Total 240π ≈ 753.6 cm³.

Rule 2: Tent: cone on a cylinder — Cylinder part first, then the cone.

  • Easy: r = 7, cylinder H = 10: 154 × 10 = 1540.
  • Medium: Cone h = 6: 1/3 × 154 × 6 = 308.
  • Exam-type: Air inside = 1848 m³.
Remember

Check that each part uses the same radius.

Common mistake

Using the slant height l instead of h in ⅓πr²h.

Memory clue

"Volume uses straight height, area uses slant."

Practice questions

Easy
  1. 1A toy is a cone of base radius 6 cm and height 19 cm placed on a hemisphere of the same radius. Find its volume (in cm³, in terms of π).
  2. 2A toy is a cone of base radius 1 cm and height 5 cm placed on a hemisphere of the same radius. Find its volume (in cm³, in terms of π).
  3. 3A toy is a cone surmounted on a hemisphere of radius 9 cm. The total height of the toy is 30 cm. Find its volume (in cm³, in terms of π).
Medium
  1. 4A toy is a cone surmounted on a hemisphere of radius 12 cm. The total height of the toy is 18 cm. Find its volume (in cm³, in terms of π).
  2. 5A tent is a cylinder of radius 6 m and height 4 m surmounted by a cone of the same radius and height 18 m. Find the volume of air inside (in m³, in terms of π).
  3. 6A tent is a cylinder of radius 12 m and height 5 m surmounted by a cone of the same radius and height 17 m. Find the volume of air inside (in m³, in terms of π).
Challenging
  1. 7A cone of the greatest size is carved out of a solid cylinder of radius 10 cm and height 16 cm. Find the volume of the wood wasted (in cm³, in terms of π).
  2. 8A toy is a cone of height 30 cm placed on a hemisphere, both with radius 10 cm. Find the ratio of the volume of the cone to the volume of the hemisphere.
Show answers (try them yourself first)
  1. 1372π
  2. 27π/3
  3. 31053π
  4. 41440π
  5. 5360π
  6. 61536π
  7. 73200π/3
  8. 83 : 2

Melting and recasting

From daily life

When a gold ring is melted into a bar, the gold does not grow or shrink: the volume stays the same.

Simple definition

When a solid is melted and recast into another shape, the volume does not change. So volume of the old solid = volume of the new solid(s). If n equal new solids are made, then n = big volume ÷ small volume.

Formula: n = big volume ÷ small volume

r = 6→27 balls, r = 2recastvolume stays samen = 216 ÷ 8= 27

In the figure: A sphere of r = 6 makes (6/2)³ = 27 balls of r = 2.

Rule 1: Many small solids — Divide the volumes; the 4/3 π cancels.

  • Easy: Sphere r = 6: 288π. Small ball r = 2: 32π/3.
  • Medium: n = 288π ÷ (32π/3) = 27.
  • Exam-type: Shortcut for spheres: n = (R/r)³ = 3³ = 27.

Rule 2: Sphere into a wire — A wire is a very long thin cylinder.

  • Easy: Sphere r = 3: 36π cm³.
  • Medium: Wire radius 0.1 cm: π × 0.01 × L = 36π, so L = 3600 cm.
  • Exam-type: The wire is 36 m long.
Remember

Write “old volume = new volume” first; the rest is algebra.

Common mistake

Equating surface areas instead of volumes.

Memory clue

"Melt it, volume is the same."

Practice questions

Easy
  1. 1A solid cylinder of radius 5 cm and height 27 cm is melted and recast into a solid cylinder of radius 3 cm. Find the height of the new cylinder (in cm).
  2. 2A solid cylinder of radius 8 cm and height 10 cm is melted and recast into a solid cylinder of radius 2 cm. Find the height of the new cylinder (in cm).
  3. 3A solid cylinder of radius 9 cm and height 16 cm is melted and recast into a solid cylinder of radius 12 cm. Find the height of the new cylinder (in cm).
Medium
  1. 4A solid metal cylinder of radius 8 cm and height 40 cm is melted and recast into spheres of radius 4 cm. How many spheres are formed?
  2. 5A solid metal cylinder of radius 12 cm and height 80 cm is melted and recast into spheres of radius 6 cm. How many spheres are formed?
  3. 6A solid cylinder of radius 7 cm and height 19 cm is melted and recast into solid cones of the same radius and height 19 cm. How many cones are formed?
Challenging
  1. 7A solid sphere of radius 10 cm is melted and recast into 25 identical solid cones of base radius 2 cm. Find the height of each cone (in cm).
  2. 8Three solid metal spheres of radii 6 cm, 8 cm and 10 cm are melted to form one solid sphere. Find the radius of the new sphere (in cm).
Show answers (try them yourself first)
  1. 175
  2. 2160
  3. 39
  4. 430
  5. 540
  6. 63
  7. 740
  8. 812

Cavities, displacement and applications

From daily life

Dropping a stone into a glass of water raises the level; the water displaced has the same volume as the stone.

Simple definition

A cavity is empty space: volume = outer solid − cavity. A solid fully dipped in water displaces water equal to its own volume, so rise in level × base area = volume of the solid. A hollow sphere has volume 4/3 π (R³ − r³).

Formula: rise × base area = volume

rise= 1r = 6 cylinderball r = 3volume 36ππ × 36 × h = 36πh = 1

In the figure: A ball r = 3 in a cylinder of r = 6 raises the water by 36π ÷ 36π = 1 cm.

Rule 1: Cavity and hollow shell — Subtract the empty part.

  • Easy: Cylinder r = 3, h = 10 with a hemispherical hole: 90π − 18π = 72π ≈ 226.1 cm³ (π = 3.14).
  • Medium: Hollow sphere R = 6, r = 3: 4/3 π (216 − 27) = 252π ≈ 791.3 cm³.
  • Exam-type: Subtract volumes, not radii.

Rule 2: Water displaced — Volume of the object = base area × rise.

  • Easy: Ball r = 3: volume 36π.
  • Medium: Cylinder r = 6: base area 36π, so rise = 1 cm.
  • Exam-type: Twelve such balls would raise it by 12 cm only if the tank is tall enough.
Remember

Ask: is the volume I need inside, outside, or the difference?

Common mistake

Using the rise in level as the volume instead of rise × base area.

Memory clue

"Outer minus empty; rise times base."

Practice questions

Easy
  1. 1A solid sphere of radius 6 cm is dropped into a cylindrical vessel of radius 12 cm that holds water. The sphere is fully submerged. By how much does the water level rise (in cm)?
  2. 2A conical cavity of the same base radius and the same height is hollowed out from a solid cylinder of radius 10 cm and height 30 cm. Find the volume of the remaining solid (in cm³, in terms of π).
  3. 3A conical cavity of the same base radius and the same height is hollowed out from a solid cylinder of radius 9 cm and height 7 cm. Find the volume of the remaining solid (in cm³, in terms of π).
Medium
  1. 4A wooden article is made by scooping out a hemisphere of radius 4 cm from each end of a solid cylinder of height 26 cm and base radius 4 cm. Find the volume of wood left (in cm³, in terms of π).
  2. 572 marbles, each a sphere of radius 3 cm, are dropped into a cylindrical jar of radius 12 cm containing water, and all get submerged. Find the rise in the water level (in cm).
  3. 6A wooden article is made by scooping out a hemisphere of radius 7 cm from each end of a solid cylinder of height 14 cm and base radius 7 cm. Find the volume of wood left (in cm³, in terms of π).
Challenging
  1. 7A wooden article is made by scooping out a hemisphere of radius 9 cm from each end of a solid cylinder of height 33 cm and base radius 9 cm. Find the volume of wood left (in cm³, in terms of π).
  2. 8A conical cavity of base radius 7 cm and height 17 cm is hollowed out from a solid cylinder of the same base and height. Find the ratio of the volume of the cavity to the volume of the remaining solid.
Show answers (try them yourself first)
  1. 12
  2. 22000π
  3. 3378π
  4. 4992π/3
  5. 518
  6. 6686π/3
  7. 71701π
  8. 81 : 2