Complete notes with formulas, concepts, and solved problems
Three-dimensional (3D) shapes have length, width, and height. Understanding their surface areas and volumes is essential for real-world applications in architecture, engineering, and everyday life.
Surface Area: Total area of all surfaces of a 3D shape (measured in square units).
Volume: Amount of space occupied by a 3D shape (measured in cubic units).
Lateral/Curved Surface Area: Area of all surfaces except the top and bottom bases.
Total Surface Area: Area of all surfaces including bases.
A cuboid is a 3D shape with 6 rectangular faces, 12 edges, and 8 vertices.
It has three dimensions: length (l), breadth (b), and height (h).
All angles are right angles (90°).
Lateral Surface Area (LSA): 2h(l + b)
Total Surface Area (TSA): 2(lb + bh + hl)
Volume: l × b × h
Diagonal: √(l² + b² + h²)
Q: A cuboid has dimensions 5 cm × 3 cm × 4 cm. Find TSA and volume.
Solution:
l = 5 cm, b = 3 cm, h = 4 cm
TSA: 2(lb + bh + hl)
= 2(5×3 + 3×4 + 4×5)
= 2(15 + 12 + 20)
= 2(47) = 94 cm²
Volume: l × b × h = 5 × 3 × 4 = 60 cm³
A cube is a special cuboid where all edges are equal (l = b = h = a).
It has 6 square faces, 12 equal edges, and 8 vertices.
All faces are congruent squares.
Lateral Surface Area: 4a²
Total Surface Area: 6a²
Volume: a³
Diagonal: a√3
where a = side length
Q: A cube has side 7 cm. Find its surface area and volume.
Solution:
a = 7 cm
TSA: 6a² = 6 × 7² = 6 × 49 = 294 cm²
Volume: a³ = 7³ = 343 cm³
A cylinder has two parallel circular bases connected by a curved surface.
Parameters: radius (r) and height (h).
Examples: pipes, cans, drums.
Curved Surface Area (CSA): 2πrh
Total Surface Area (TSA): 2πr(r + h)
Or: 2πr² + 2πrh (area of both circles + curved surface)
Volume: πr²h
Q: A cylinder has radius 7 cm and height 10 cm. Find CSA, TSA, and volume. (Use π = 22/7)
Solution:
r = 7 cm, h = 10 cm
CSA: 2πrh = 2 × (22/7) × 7 × 10 = 440 cm²
TSA: 2πr(r + h) = 2 × (22/7) × 7 × (7 + 10) = 44 × 17 = 748 cm²
Volume: πr²h = (22/7) × 7² × 10 = 22 × 7 × 10 = 1540 cm³
A cone has a circular base and a curved surface tapering to a point (apex).
Parameters: radius (r), height (h), slant height (l).
Relationship: l² = r² + h² (Pythagoras theorem)
Slant Height: l = √(r² + h²)
Curved Surface Area: πrl
Total Surface Area: πr(r + l)
Or: πr² + πrl (base area + curved surface)
Volume: (1/3)πr²h
Q: A cone has radius 5 cm and height 12 cm. Find slant height, CSA, and volume.
Solution:
r = 5 cm, h = 12 cm
Slant height: l = √(r² + h²) = √(25 + 144) = √169 = 13 cm
CSA: πrl = (22/7) × 5 × 13 ≈ 204.3 cm²
Volume: (1/3)πr²h = (1/3) × (22/7) × 25 × 12 ≈ 314.3 cm³
A sphere is a perfectly round 3D shape where all points on surface are equidistant from centre.
Parameter: radius (r).
Examples: ball, globe, planets.
Surface Area: 4πr²
Volume: (4/3)πr³
Note: Sphere has only one surface, no bases.
A hemisphere is exactly half of a sphere.
It has a curved surface and a circular flat base.
Curved Surface Area: 2πr²
Total Surface Area: 3πr²
(CSA + base area = 2πr² + πr² = 3πr²)
Volume: (2/3)πr³
Q: A sphere has radius 6 cm. Find its surface area and volume. Also find volume of hemisphere with same radius.
Solution:
r = 6 cm
Sphere Surface Area: 4πr² = 4 × (22/7) × 36 ≈ 452.4 cm²
Sphere Volume: (4/3)πr³ = (4/3) × (22/7) × 216 ≈ 904.8 cm³
Hemisphere Volume: (2/3)πr³ = (2/3) × (22/7) × 216 ≈ 452.4 cm³
Note: Hemisphere volume = ½ × Sphere volume
| Shape | TSA Formula | Volume Formula |
|---|---|---|
| Cuboid | 2(lb + bh + hl) | l × b × h |
| Cube | 6a² | a³ |
| Cylinder | 2πr(r + h) | πr²h |
| Cone | πr(r + l) | (1/3)πr²h |
| Sphere | 4πr² | (4/3)πr³ |
| Hemisphere | 3πr² | (2/3)πr³ |
TSA = 2(lb + bh + hl)
LSA = 2h(l + b)
Volume = l × b × h
Rectangular box
TSA = 6a²
LSA = 4a²
Volume = a³
All sides equal
CSA = 2πrh
TSA = 2πr(r + h)
Volume = πr²h
Circular bases
Slant height: l = √(r²+h²)
CSA = πrl
TSA = πr(r + l)
Volume = (1/3)πr²h
Surface Area = 4πr²
Volume = (4/3)πr³
Perfectly round
CSA = 2πr²
TSA = 3πr²
Volume = (2/3)πr³
Half sphere
Cone volume = 1/3 × Cylinder volume
Hemisphere volume = 1/2 × Sphere volume
l² = r² + h² (cone slant height)
Area → cm², m² (square units)
Volume → cm³, m³ (cubic units)
Always use π = 22/7 or 3.14
• Memorize ALL formulas - make flashcards for each shape!
• Remember: Volume always has cubic units (cm³), Surface area has square units (cm²).
• For cylinder and cone, distinguish between CSA (curved only) and TSA (includes bases).
• Cone volume = 1/3 of cylinder volume (same base and height).
• Always calculate slant height first for cone problems.
• Hemisphere = Half sphere, but TSA includes flat circular base too!
• Practice unit conversions: 1 m³ = 1,000,000 cm³, 1 litre = 1000 cm³.
• Draw diagrams and label all given measurements clearly.
Q1: A cuboid-shaped room is 5m × 4m × 3m. Find cost of painting walls at ₹10/m².
Solution: LSA = 2h(l+b) = 2×3×(5+4) = 54 m². Cost = 54×10 = ₹540
Q2: A cylindrical tank has diameter 14 m and height 10 m. Find capacity in litres.
Solution: r = 7 m, Volume = πr²h = (22/7)×49×10 = 1540 m³ = 1,540,000 litres
Q3: Compare: Volume of sphere vs hemisphere with radius 3 cm.
Solution: Sphere = (4/3)π(27) = 36π cm³, Hemisphere = (2/3)π(27) = 18π cm³
Hemisphere = ½ × Sphere (as expected!)