Mathematics Classes VI-X for School Assistants and TET Paper 2A · Chapter 10

Mensuration: Cylinder, Cone, Sphere and Combined Solids

What to remember

Sphere: surface area 4πr², volume (4/3)πr³. Cone: CSA πrl, volume (1/3)πr²h, with l² = r² + h². Hemisphere: CSA 2πr², TSA 3πr², volume (2/3)πr³.
When a solid is melted and recast, its volume stays the same; its surface area usually changes.
For combined solids add the volumes, but add only the exposed surfaces to find the total surface area.

Units and basics

Area is measured in square units and volume in cubic units. 1 m = 100 cm, so 1 m³ = 10,00,000 cm³. 1 litre = 1000 cm³ and 1 m³ = 1000 litres. Unless told otherwise, use π = 22/7 when the radius or the answer works with 7, and leave π in the answer if the question asks for it.

Curved surface area (CSA) or lateral area is the area of the curved face only. Total surface area (TSA) includes flat faces. Volume is the space occupied (capacity if the solid is hollow).

Cuboid, cube and cylinder

Solid	CSA / lateral	TSA	Volume
Cuboid l × b × h	2h(l + b)	2(lb + bh + hl)	lbh
Cube a	4a²	6a²	a³
Cylinder r, h	2πrh	2πr(r + h)	πr²h

Diagonal of a cuboid = √(l² + b² + h²); diagonal of a cube = a√3. Hollow cylinder with outer radius R and inner radius r: volume = πh(R² − r²).

Example: a cuboid tank 2 m × 1.5 m × 1 m holds 3 m³ = 3000 litres. A cube with TSA 150 cm² has a = 5 cm and volume 125 cm³. A cylinder with r = 7 cm and h = 10 cm has volume 1540 cm³ and CSA 440 cm².

Cone

For a cone with base radius r, height h and slant height l, l = √(r² + h²).

Quantity	Formula
Curved surface area	πrl
Total surface area	πr(l + r)
Volume	(1/3)πr²h

A cone has one-third of the volume of a cylinder with the same base and height. When a cone is cut open, the curved surface becomes a sector of a circle of radius l whose arc length equals the base circumference 2πr. The cone from a semicircle of radius R has slant height R and base radius R/2.

Example: r = 3 and h = 4 give l = 5, CSA = 15π and volume = 12π. For r = 7 and h = 24, l = 25 and CSA = 22/7 × 7 × 25 = 550.

Sphere and hemisphere

Solid	CSA	TSA	Volume
Sphere r	–	4πr²	(4/3)πr³
Hemisphere r	2πr²	3πr²	(2/3)πr³
Spherical shell R, r	–	–	(4/3)π(R³ − r³)

A sphere has no edges or flat faces. Its volume varies as r³ and its surface area as r². If the radius is doubled, the volume becomes 8 times and the surface area 4 times. The largest sphere inside a cube of side a has radius a/2. The largest cube inside a sphere has space diagonal equal to the diameter.

Examples: r = 7 gives surface area 616. A sphere of radius 3 has volume 36π. A hemisphere of radius 6 has volume 144π.

Frustum of a cone

A frustum is what remains of a cone when a smaller cone is cut off by a plane parallel to the base. With radii R and r, height h and slant height l:

Quantity	Formula
Slant height	l = √(h² + (R − r)²)
Volume	(πh/3)(R² + Rr + r²)
CSA	π(R + r)l
TSA	π(R + r)l + πR² + πr²

Example: R = 7, r = 4, h = 4 gives l = 5 and CSA = 55π. R = 4, r = 2, h = 3 gives volume 28π.

Combined solids and conversion

For combined solids (a cone on a hemisphere, a cylinder with hemispherical ends, a cylinder with a cone on top), draw the figure and list each part.

Volume: add the volumes of the parts (or subtract if one part is scooped out).
Surface area: add the curved areas of the exposed parts. Where two solids join, the joined face is hidden and is left out.

Worked examples:

1. A cone (r = 3, h = 4, l = 5) on a hemisphere (r = 3). Volume = 12π + 18π = 30π. Surface = 15π + 18π = 33π.
2. A capsule: cylinder r = 3, h = 10, with a hemisphere on each end. Surface = 2π(3)(10) + 4π(9) = 96π. Volume = 90π + 36π = 126π.
3. A sphere of radius 6 recast into spheres of radius 2: number = 6³/2³ = 27.
4. A cylinder with r = 3, h = 8 recast into spheres of radius 3: 72π/36π = 2 spheres.
5. A sphere of radius 3 recast into a cone of base radius 3: 36π = (1/3)π(9)h gives h = 12.

For conversion questions write "volume of old solid = volume of new solid (or n × volume of one)" and cancel π. If a vessel is filled or water is poured, equate volumes; if a cylinder's water level rises when a sphere is dropped in, rise × base area = volume of the sphere.

Practical problems

Water in a tank. Capacity of a cylindrical tank = πr²h. If a pipe of cross-section radius r₁ delivers water at speed v, the volume per unit time is πr₁²v. Time to fill = capacity ÷ rate, so keep units consistent (metres with metres, then convert cubic metres to litres).

Ice-cream cone. A cone of radius r and height h filled with ice cream and topped by a hemisphere of the same radius has volume (1/3)πr²h + (2/3)πr³.

Hollow bowl. A hemispherical bowl of inner radius r holds (2/3)πr³; the metal used for a bowl of thickness t is the difference of the outer and inner hemispherical volumes.

Sector to cone. For a sector of radius l and angle θ made into a cone, the arc length (θ/360) × 2πl equals 2πr, so r = lθ/360. The slant height of the cone is l.

Rise in water level. When a solid is completely dipped into a cylindrical vessel, the volume of water displaced equals the volume of the solid, so the rise in level = volume of solid ÷ base area of the vessel.

Pipe or well. Earth dug out from a cylindrical well of radius r and depth h has volume πr²h; if it is spread evenly over a rectangle, thickness = volume ÷ area.

Check list for every problem: (1) draw the figure, (2) write the formula, (3) substitute with units, (4) check that the answer is reasonable (for example, a cone's volume should be smaller than the cylinder that encloses it).

Classroom angle

Teach with real objects: fill a cone with sand and pour it into a cylinder of the same base and height to show the one-third relation, open out a paper cone to see the sector, and use clay to show that recasting keeps the volume. Common errors are using h instead of l in the cone area, mixing up CSA and TSA, adding hidden faces of combined solids, forgetting to convert units, and applying 22/7 when the radius is not a multiple of 7.

Exam traps

Use the slant height l for the cone's CSA, but the height h for its volume.
The sphere's surface area is 4πr², while the hemisphere's CSA is 2πr² and TSA is 3πr².
A solid hemisphere's TSA includes the flat circular face; a hollow bowl does not.
Melting and recasting conserves volume, not surface area.
Doubling the radius gives 4 times the area and 8 times the volume of a sphere.
1 litre is 1000 cm³, not 100 cm³; 1 m³ is 1000 litres.
In combined solids, do not count the joined faces in the surface area.
The frustum's slant height uses the difference of the radii, not their sum.

One-liners

1. Sphere: 4πr² and (4/3)πr³.
2. Cone: πrl and (1/3)πr²h.
3. Cylinder: 2πrh and πr²h.
4. Hemisphere: 2πr², 3πr² and (2/3)πr³.
5. Slant height l = √(r² + h²).
6. A cone is one-third of the cylinder with the same base and height.
7. 1 litre = 1000 cm³.
8. 1 m³ = 1000 litres.
9. A cuboid's diagonal is √(l² + b² + h²).
10. A cube's diagonal is a√3.
11. A frustum's volume is (πh/3)(R² + Rr + r²).
12. A shell's volume is (4/3)π(R³ − r³).

Practice questions

The surface area of a sphere of radius r is
1. πr²
2. 4πr²
3. 2πr²
4. 3πr²
Answer
B. 4πr²
Standard formula.
The volume of a sphere of radius r is
1. (1/3)πr³
2. (2/3)πr³
3. 4πr²
4. (4/3)πr³
Answer
D. (4/3)πr³
Standard formula.
The volume of a cone with base radius r and height h is
1. πr²h
2. (1/3)πr²h
3. (2/3)πr²h
4. (1/2)πr²h
Answer
B. (1/3)πr²h
A cone is one-third of the cylinder with the same base and height.
The curved surface area of a cone with base radius r and slant height l is
1. πrl
2. πrh
3. πr²
4. 2πrl
Answer
A. πrl
CSA = πrl.
The slant height of a cone with radius r and height h is
1. √(r² + h²)
2. r² + h²
3. r + h
4. √(h² − r²)
Answer
A. √(r² + h²)
The slant height is the hypotenuse of the right triangle formed.
The curved surface area of a hemisphere of radius r is
1. 3πr²
2. πr²
3. 2πr²
4. 4πr²
Answer
C. 2πr²
Half of the sphere's 4πr².
The total surface area of a solid hemisphere of radius r is
1. 4πr²
2. 3πr²
3. 2πr²
4. 5πr²
Answer
B. 3πr²
2πr² curved + πr² flat base.
The volume of a hemisphere of radius r is
1. (1/3)πr³
2. (4/3)πr³
3. 2πr³
4. (2/3)πr³
Answer
D. (2/3)πr³
Half of the sphere's volume.
The total surface area of a closed cylinder is
1. 2πr(r + h)
2. πr(r + h)
3. πr²h
4. 2πrh
Answer
A. 2πr(r + h)
Curved area 2πrh plus two circular ends 2πr².
The ratio of the volume of a cone to the volume of a cylinder with the same base radius and height is
1. 1 : 1
2. 1 : 3
3. 2 : 3
4. 1 : 2
Answer
B. 1 : 3
Cone = (1/3) cylinder.
The surface area of a sphere of radius 7 cm (take π = 22/7) is
1. 1232 cm²
2. 154 cm²
3. 616 cm²
4. 308 cm²
Answer
C. 616 cm²
4 × 22/7 × 49 = 616.
The volume of a sphere of radius 21 cm (take π = 22/7) is
1. 19404 cm³
2. 7392 cm³
3. 58212 cm³
4. 38808 cm³
Answer
D. 38808 cm³
(4/3) × 22/7 × 9261 = 38808.
The volume of a sphere of radius 3 cm is
1. 27π cm³
2. 108π cm³
3. 12π cm³
4. 36π cm³
Answer
D. 36π cm³
(4/3)π × 27 = 36π.
The total surface area of a solid hemisphere of radius 7 cm (π = 22/7) is
1. 462 cm²
2. 308 cm²
3. 154 cm²
4. 616 cm²
Answer
A. 462 cm²
3πr² = 3 × 22/7 × 49 = 462.
The volume of a cone with radius 3 cm and height 4 cm is
1. 36π cm³
2. 12π cm³
3. 24π cm³
4. 4π cm³
Answer
B. 12π cm³
(1/3)π × 9 × 4 = 12π.
The curved surface area of a cone with radius 3 cm and height 4 cm is
1. 12π cm²
2. 9π cm²
3. 15π cm²
4. 24π cm²
Answer
C. 15π cm²
Slant height 5, so πrl = 15π.
A cone has radius 7 cm and height 24 cm. Its curved surface area (π = 22/7) is
1. 550 cm²
2. 154 cm²
3. 1100 cm²
4. 528 cm²
Answer
A. 550 cm²
l = 25; π r l = 22/7 × 7 × 25 = 550.
A cone has radius 6 cm and height 7 cm. Its volume (π = 22/7) is
1. 132 cm³
2. 792 cm³
3. 264 cm³
4. 528 cm³
Answer
C. 264 cm³
(1/3) × 22/7 × 36 × 7 = 264.
A cylinder has radius 7 cm and height 10 cm. Its volume (π = 22/7) is
1. 770 cm³
2. 440 cm³
3. 3080 cm³
4. 1540 cm³
Answer
D. 1540 cm³
22/7 × 49 × 10 = 1540.
A cylinder has radius 7 cm and height 10 cm. Its curved surface area (π = 22/7) is
1. 748 cm²
2. 440 cm²
3. 1540 cm²
4. 220 cm²
Answer
B. 440 cm²
2 × 22/7 × 7 × 10 = 440.
The volume of a hemisphere of radius 6 cm is
1. 288π cm³
2. 72π cm³
3. 216π cm³
4. 144π cm³
Answer
D. 144π cm³
(2/3)π × 216 = 144π.
If the radius of a sphere is doubled, its volume becomes
1. 2 times
2. 6 times
3. 8 times
4. 4 times
Answer
C. 8 times
Volume varies as r³.
If the radius of a sphere is doubled, its surface area becomes
1. 8 times
2. 4 times
3. 2 times
4. 6 times
Answer
B. 4 times
Surface area varies as r².
If the radius of a cone is doubled and its height is kept the same, its volume becomes
1. 4 times
2. 6 times
3. 2 times
4. 8 times
Answer
A. 4 times
Volume varies as r² when h is constant.
A solid sphere of radius 6 cm is melted and recast into small spheres of radius 2 cm. The number of small spheres is
1. 9
2. 3
3. 27
4. 18
Answer
C. 27
216/8 = 27.
A solid cylinder of radius 3 cm and height 8 cm is melted into spheres of radius 3 cm. The number of spheres is
1. 2
2. 1
3. 3
4. 4
Answer
A. 2
72π / 36π = 2.
A solid sphere of radius 3 cm is melted and recast into a cone of base radius 3 cm. The height of the cone is
1. 6 cm
2. 4 cm
3. 9 cm
4. 12 cm
Answer
D. 12 cm
36π = (1/3)π × 9 × h gives h = 12.
The largest sphere that fits inside a cube of side 6 cm has volume
1. 288π cm³
2. 36π cm³
3. 12π cm³
4. 72π cm³
Answer
B. 36π cm³
The radius is 3 cm, so (4/3)π × 27 = 36π.
A cone is made from a semicircular sheet of radius 14 cm. The radius of the cone's base is
1. 7 cm
2. 3.5 cm
3. 28 cm
4. 14 cm
Answer
A. 7 cm
The arc length πl = 2πr gives r = l/2 = 7.
A toy is a cone (radius 3 cm, height 4 cm) joined to a hemisphere of radius 3 cm on its base. Its volume is
1. 42π cm³
2. 12π cm³
3. 18π cm³
4. 30π cm³
Answer
D. 30π cm³
Cone 12π + hemisphere 18π = 30π.
For the same toy (cone of slant height 5 cm on a hemisphere of radius 3 cm) the total surface area is
1. 48π cm²
2. 33π cm²
3. 15π cm²
4. 18π cm²
Answer
B. 33π cm²
Cone CSA 15π + hemisphere CSA 18π = 33π.
A capsule is a cylinder of radius 3 cm and height 10 cm with a hemisphere on each end. Its total surface area is
1. 66π cm²
2. 126π cm²
3. 96π cm²
4. 60π cm²
Answer
C. 96π cm²
2π × 3 × 10 + 4π × 9 = 60π + 36π = 96π.
A frustum has radii 7 cm and 4 cm and height 4 cm. Its slant height is
1. 4 cm
2. √65 cm
3. 5 cm
4. 3 cm
Answer
C. 5 cm
l = √(4² + 3²) = 5.
The volume of a frustum with radii 4 cm and 2 cm and height 3 cm is
1. 84π cm³
2. 28π cm³
3. 14π cm³
4. 56π cm³
Answer
B. 28π cm³
(πh/3)(R² + Rr + r²) = π(16 + 8 + 4) = 28π.
A cuboid tank measures 2 m × 1.5 m × 1 m. Its capacity in litres is
1. 30000
2. 300
3. 30
4. 3000
Answer
D. 3000
3 m³ = 3000 litres.
A cuboid has sides 3, 4 and 12 units. Its space diagonal is
1. 13 units
2. 12 units
3. 5 units
4. 19 units
Answer
A. 13 units
√(9 + 16 + 144) = 13.
A cube has total surface area 150 cm². Its volume is
1. 216 cm³
2. 25 cm³
3. 150 cm³
4. 125 cm³
Answer
D. 125 cm³
6a² = 150 gives a = 5, so V = 125.
Consider the statements: 1. A cone's volume is one-third of the volume of a cylinder with the same base and height. 2. The surface area of a sphere is 2πr². Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
The sphere's surface area is 4πr².
Consider the statements: 1. When a solid is melted and recast, its volume stays the same. 2. When a solid is melted and recast, its surface area stays the same. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Volume is conserved, but surface area generally changes.
Consider the statements: 1. The curved surface area of a hemisphere is 2πr². 2. The total surface area of a solid hemisphere is 3πr². Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are right.
Consider the statements: 1. The curved surface area of a cone is πrh. 2. The curved surface area of a cone is πrl, with l the slant height. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
The cone formula uses the slant height l, not the height h.
Consider the statements: 1. 1 litre equals 100 cm³. 2. The total surface area of a closed cylinder is 2πrh. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
1 litre = 1000 cm³; the cylinder's total area is 2πr(r + h).
Which pair is correctly matched?
1. Cylinder : volume 2πrh
2. Cone : volume (1/3)πr²h
3. Sphere : surface area (4/3)πr³
4. Hemisphere : volume (4/3)πr³
Answer
B. Cone : volume (1/3)πr²h
Only the cone pair is right.
A teacher wants learners to see why a cone's volume is one-third of a cylinder's. The best activity is to
1. fill a cone with sand or water and pour it into a cylinder of the same base and height three times
2. state the formula and move on
3. draw the cone only on the board
4. ask learners to memorise 1/3
Answer
A. fill a cone with sand or water and pour it into a cylinder of the same base and height three times
Practical filling gives meaning to the 1/3 factor.
A learner uses height h instead of slant height l for the curved surface area of a cone. The best way to correct this is to
1. tell the learner to be careful
2. ignore the error since the volume formula uses h
3. show a paper cone opened out to a sector so that the slant edge is the sector radius
4. give a longer list of formulas
Answer
C. show a paper cone opened out to a sector so that the slant edge is the sector radius
Opening out a cone makes the role of slant height visible.

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Chapter quiz · 10 questions

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अध्याय पढ़ने के बाद अभ्यास करें।

1.Find the area of a triangle with base 8 m and height 6 m.
2.Find the area of a circle with radius 7 m. (Use pi = 22/7)
3.Find the area of a rectangle with length 15 m and breadth 7 m.
4.Find the area of a square with side 5 m.
5.Find the area of a rectangle with length 10 m and breadth 9 m.
6.Find the area of a rectangle with length 20 m and breadth 9 m.
7.Find the area of a triangle with base 12 m and height 6 m.
8.Find the area of a rectangle with length 12 m and breadth 6 m.
9.Find the area of a triangle with base 14 m and height 9 m.
10.Find the area of a circle with radius 21 m. (Use pi = 22/7)

10 left. You can check any time.