1. Chapter at a Glance
How much paint is needed for a water tank? How much water can it hold? The first question is about surface area — the total area of the outside of a solid. The second is about volume — the space inside it. In this chapter we develop formulas for the surface area and volume of cuboids, cubes, cylinders, cones, pyramids, spheres and hemispheres, see where they come from, and use them for real objects — including quick "guesstimates" to check whether an answer is sensible.
- Lateral (curved) surface area, total surface area and volume.
- Cuboid and cube.
- Right circular cylinder.
- Right circular cone and slant height.
- Pyramids and the "one-third" rule.
- Sphere and hemisphere.
- Units, conversions (1 m³ = 1000 L) and estimation.
2. Cuboid and Cube
A cuboid of length l, breadth b and height h has 6 rectangular faces in 3 equal pairs.
| Solid | Lateral surface area | Total surface area | Volume |
|---|---|---|---|
| Cuboid | 2h(l + b) | 2(lb + bh + hl) | lbh |
| Cube (edge a) | 4a² | 6a² | a³ |
Lateral area = area of the four walls (no top or bottom) — useful for painting the walls of a room.
Diagonal of a cuboid = √(l² + b² + h²); of a cube = a√3.
A room is 6 m × 4 m × 3 m. Find the cost of painting its four walls and ceiling at ₹25/m².
Solution: Walls = 2 × 3 × (6 + 4) = 60 m²; ceiling = 24 m². Total 84 m² → ₹2,100.
3. Cylinder
Unroll the curved surface of a cylinder (radius r, height h): it becomes a rectangle of length 2πr (the circumference) and width h.
| Quantity | Formula |
|---|---|
| Curved surface area (CSA) | 2πrh |
| Total surface area (TSA) | 2πr(r + h) |
| Volume | πr²h (base area × height) |
| Hollow cylinder volume | πh(R² − r²) |
A cylindrical tank has radius 1.4 m and height 2 m. How many litres does it hold? (π = 22/7)
Solution: V = (22/7) × 1.96 × 2 = 12.32 m³ = 12,320 L.
4. Cone
A right circular cone has base radius r, height h and slant height l, where l = √(r² + h²) (Baudhāyana–Pythagoras).
| Quantity | Formula |
|---|---|
| Curved surface area | πrl |
| Total surface area | πr(l + r) |
| Volume | ⅓πr²h |
Why ⅓? Fill a cone with sand or water and pour it into a cylinder with the same base and height: it takes exactly three cones to fill the cylinder.
Why πrl? Cut the curved surface along a slant line and flatten it: it becomes a sector of a circle of radius l with arc length 2πr. Its area = ½ × arc × radius = ½ × 2πr × l = πrl.
A cone has radius 7 cm and height 24 cm. Find its slant height, CSA and volume.
Solution: l = √(49 + 576) = 25 cm. CSA = (22/7) × 7 × 25 = 550 cm². V = ⅓ × (22/7) × 49 × 24 = 1232 cm³.
5. Pyramid
A pyramid has a polygon base and triangular faces meeting at the apex. Like the cone:
Volume = ⅓ × base area × height. Surface area = base area + areas of the triangular faces.
A cube can be cut into 3 identical square pyramids that meet at one corner — a neat way to see the ⅓.
A square pyramid has base side 10 m and height 12 m. Find its volume and total surface area.
Solution: V = ⅓ × 100 × 12 = 400 m³. Slant height of each face = √(12² + 5²) = 13. Four triangles: 4 × ½ × 10 × 13 = 260. TSA = 260 + 100 = 360 m².
6. Sphere and Hemisphere
| Solid | Surface area | Volume |
|---|---|---|
| Sphere | 4πr² | (4/3)πr³ |
| Hemisphere | CSA 2πr², TSA 3πr² | (2/3)πr³ |
| Spherical shell | — | (4/3)π(R³ − r³) |
Archimedes' discovery: a sphere fits exactly inside a cylinder of the same radius and height 2r. The sphere's surface area equals the cylinder's curved surface (2πr × 2r = 4πr²), and its volume is two-thirds of the cylinder's (⅔ × πr² × 2r = (4/3)πr³). Archimedes was so proud of this that he wanted it carved on his tomb.
Cone : hemisphere : cylinder of the same radius r and height r have volumes in ratio 1 : 2 : 3.
Find the surface area and volume of a ball of radius 21 cm.
Solution: SA = 4 × (22/7) × 441 = 5544 cm². V = (4/3) × (22/7) × 9261 = 38808 cm³.
7. Units and Guesstimates
| Conversion | Value |
|---|---|
| 1 cm³ | 1 mL |
| 1 L | 1000 cm³ |
| 1 m³ | 1000 L = 10⁶ cm³ |
Guesstimate: about how much water does a 2-litre bottle hold compared with a bucket? Model the bucket as a cylinder of radius ~15 cm and height ~30 cm: π × 225 × 30 ≈ 21,000 cm³ ≈ 21 L — so roughly 10 bottles. Quick estimates like this help catch calculation mistakes (e.g. a cup "holding 30 L").
Scaling: if every length is multiplied by k, surface area is multiplied by k² and volume by k³. A ball with double the radius needs 4 times the leather and holds 8 times the air.
8. Solved Problems (Board Pattern)
A metal sphere of radius 6 cm is melted and recast into a cylindrical wire of radius 0.2 cm. Find the length of the wire.
Solution: (4/3)π(216) = π(0.04)L → 288 = 0.04L → L = 7200 cm = 72 m.
A conical tent is 10 m high with base radius 24 m. Find the canvas needed (π = 3.14).
Solution: l = √(576 + 100) = 26 m. CSA = 3.14 × 24 × 26 = 1959.36 m².
A hemispherical bowl has inner radius 10.5 cm. How much soup can it hold? (π = 22/7)
Solution: (2/3) × (22/7) × 1157.625 = 2425.5 cm³ ≈ 2.43 L.
9. Master Formula Table
| Solid | CSA / LSA | TSA | Volume |
|---|---|---|---|
| Cuboid | 2h(l + b) | 2(lb + bh + hl) | lbh |
| Cube | 4a² | 6a² | a³ |
| Cylinder | 2πrh | 2πr(r + h) | πr²h |
| Cone | πrl | πr(l + r) | ⅓πr²h |
| Pyramid | sum of triangles | + base | ⅓ × base × h |
| Sphere | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere | 2πr² | 3πr² | (2/3)πr³ |
10. Master Summary Table
| Idea | Key fact | Exam focus |
|---|---|---|
| Prisms (cuboid, cylinder) | Volume = base area × height | Tanks, rooms |
| Pointed solids (cone, pyramid) | Volume = ⅓ base × height | Tents, heaps |
| Slant height | l = √(r² + h²) | Canvas, cone CSA |
| Sphere | Archimedes: ⅔ of cylinder | Balls, domes |
| Recasting | Volume stays the same | Melting problems |
| Scaling | Area × k², volume × k³ | Compare sizes |
Chart — volumes with r = h = 3 cm (π ≈ 3.14):
11. Mind Map
- Cuboid, cube
- Walls = 2h(l + b)
- Cylinder 2πrh
- Cone πrl
- Cone, pyramid
- 3 fill a prism
- 4πr², (4/3)πr³
- Hemisphere
12. Tips and Tricks
- Read carefully: "how much metal sheet" = surface area; "how much it holds" = volume.
- Open containers (bucket, tank without lid): subtract the top area.
- For melting/recasting, equate volumes — π often cancels.
- Using height h instead of slant height l in the cone's CSA.
- Using the diameter as the radius.
- Mixing units: convert everything to cm or m before calculating.
13. Important One-liners
- Volume of any prism or cylinder = base area × height.
- A cone's volume is one-third of the cylinder with the same base and height.
- Slant height of a cone: l = √(r² + h²).
- Surface area of a sphere = 4πr² — four times the area of its great circle.
- 1 m³ = 1000 litres.
- Doubling the radius of a sphere makes its volume 8 times bigger.
14. Exam-pattern Questions with Detailed Answers
Find the total surface area of a cube of edge 5 cm.
Show answer
6 × 25 = 150 cm².
Answer: 150 cm²
Find the volume of a cylinder with r = 7 cm and h = 10 cm (π = 22/7).
Show answer
(22/7) × 49 × 10 = 1540 cm³.
Answer: 1540 cm³
A cone and a cylinder have the same base and height. The cylinder's volume is 270 cm³. Find the cone's volume.
Show answer
⅓ × 270 = 90 cm³.
Answer: 90 cm³
Find the curved surface area of a cone with radius 5 cm and height 12 cm (π = 3.14).
Show answer
l = 13. CSA = 3.14 × 5 × 13 = 204.1 cm².
Answer: 204.1 cm²
How many litres of water does a tank 3 m × 2 m × 1.5 m hold?
Show answer
9 m³ = 9000 L.
Answer: 9000 L
The surface area of a sphere is 616 cm². Find its radius and volume (π = 22/7).
Show answer
4πr² = 616 → r² = 49 → r = 7. V = (4/3)(22/7)(343) = 1437⅓ cm³.
Answer: r = 7 cm, V ≈ 1437.33 cm³
The radius of a sphere is tripled. How many times does its volume become?
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3³ = 27 times.
Answer: 27 times
A heap of wheat is a cone of diameter 10.5 m and height 3 m. Find its volume (π = 22/7).
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r = 5.25. V = ⅓ × (22/7) × 27.5625 × 3 = 86.625 m³.
Answer: 86.625 m³
A solid toy is a cone of radius 3.5 cm and height 12 cm mounted on a hemisphere of the same radius. Find its total surface area (π = 22/7).
Show answer
l = √(12.25 + 144) = 12.5. Cone CSA = (22/7)(3.5)(12.5) = 137.5. Hemisphere CSA = 2(22/7)(12.25) = 77. Total = 214.5 cm².
Answer: 214.5 cm²
How many lead shots of radius 0.5 cm can be made from a cuboid of lead 11 cm × 10 cm × 7 cm? (π = 22/7)
Show answer
Cuboid = 770 cm³. One shot = (4/3)(22/7)(0.125) = 11/21 cm³. Number = 770 ÷ (11/21) = 1470.
Answer: 1470
A cylindrical pipe has inner radius 7 cm, and water flows through it at 5 m/s. How many litres flow out in 1 minute? (π = 22/7)
Show answer
Length of water per minute = 300 m = 30000 cm. Volume = (22/7) × 49 × 30000 = 4,620,000 cm³ = 4620 L.
Answer: 4620 litres
Show that the volumes of a cone, hemisphere and cylinder of the same radius r and height r are in the ratio 1 : 2 : 3.
Show answer
Cone: ⅓πr²·r = ⅓πr³. Hemisphere: ⅔πr³. Cylinder: πr²·r = πr³. Ratio ⅓ : ⅔ : 1 = 1 : 2 : 3. ∎
Answer: 1 : 2 : 3
15. Quick Revision Summary
- Cuboid: 2(lb + bh + hl), lbh; cube: 6a², a³.
- Cylinder: 2πrh, 2πr(r + h), πr²h.
- Cone: l = √(r² + h²), πrl, ⅓πr²h; pyramid: ⅓ × base × h.
- Sphere: 4πr², (4/3)πr³; hemisphere: 3πr² (total), (2/3)πr³.
- 1 m³ = 1000 L; area scales by k², volume by k³.