Volume
Volume - Methods to calculate volume of cube, cuboid, cylinder, cone, sphere and other solids
Learning Content
📦 What is volume?
The amount of space a three-dimensional (3D) shape occupiescapacity(Volume) is called.
capacityin cubic units(Cubic Units) are measured.
📏 Units
| unit of length | Capacitance unit |
|---|---|
| meter (m) | cubic meter (m³) |
| Centimeter (cm) | Cubic cm (cm³) |
| millimeter (mm) | Cubic mm (mm³) |
| feet (ft) | cubic feet (cu.ft / ft³) |
🔄 Unit Conversions
| Change | value |
|---|---|
| 1 m³ | 1000000 cm³ (10⁶ cm³) |
| 1 cm³ | 1000 mm³ |
| 1 liter | 1000 cm³ = 1000 ml |
| 1 m³ | 1000 litres |
| 1 Kilometer (kl) | 1000 liters = 1 m³ |
| 1 cu.ft | 28.317 liters ≈ 28.3 liters |
📊 Basic Solids
1. Cube
A solid with six square faces
All edges are equal
Capacity = a³
2. Cuboid
A solid with six rectangular faces
Opposite faces are equal
Capacity = l × b × h
3. Cylinder
A solid with two circular bases
Capacitance = πr²h
4. Cone
A round bottom, a top
Capacitance = ⅓πr²h
5. Sphere
All points are equidistant from the center
Capacitance = (4/3)πr³
6. Hemisphere
Half of the sphere
Capacitance = (2/3)πr³
📚 Key Codes
| Code | Meaning | English |
|---|---|---|
| a | The edge of the cube | Edge of Cube |
| l | length | Length |
| b | width | Breadth |
| h | height | Height |
| r | Radius | Radius |
| d | diameter | Diameter |
| l (slant) | Slope height | Slant Height |
| V | capacity | Volume |
| TSA | Total surface area | Total Surface Area |
| CSA/LSA | Broadcasting | Curved/Lateral Surface Area |
🎯 Capacitance vs Surface Area
| Volume | Surface Area |
|---|---|
| Inside Space (3D) | External Surface (2D) |
| Three-dimensional scale | Two-dimensional scale |
| cubic meter, cubic cm | Square meter, square cm. |
| Fill with water and sand | to paint |
| Can be measured in liters | In square units only |
💡 Things to remember
- π (pi)= 22/7 = 3.14159...
- 1 liter= 1000 cm³ = 1000 ml
- 1 m³= 1000 litres
- Cone capacitance= ⅓ share of cylinder
- Hemispherical capacity= ½ part of sphere
- Diameter = 2 × radius(d = 2r)
- Diagonal of cube= a√3
📐 Capacitance formulas - all solids
All Capacity & Surface Formulas for TNPSC Exam!
🔲 Cube - edge = a
| capacity | V = a³ |
| Total surface area | TSA = 6a² |
| (Page 4) | LSA = 4a² |
| Diagonal | d = a√3 |
| margin (from capacity) | a = ∛V |
| Number of edges | 12 |
| Number of faces | 6 |
| Number of vertices | 8 |
📦 Cuboid - l × b × h
| capacity | V = l × b × h |
| Total surface area | TSA = 2(lb + bh + hl) |
| (Page 4) | LSA = 2h(l + b) |
| Diagonal | d = √(l² + b² + h²) |
| Addition of margins | 4(l + b + h) |
🥫 Cylinder - radius r, height h
| capacity | V = πr²h |
| Broadcasting | CSA = 2πrh |
| Total surface area | TSA = 2πr(r + h) |
| Subsurface | πr² |
V = πh(R² - r²)
TSA = 2π(R + r)(h + R - r)
🍦 Cone - radius r, height h, slope l
| capacity | V = ⅓πr²h |
| Broadcasting | CSA = πrl |
| Total surface area | TSA = πr(r + l) |
| Slope height | l = √(r² + h²) |
| height | h = √(l² - r²) |
🌍 Sphere - radius r
| capacity | V = (4/3)πr³ |
| surface | SA = 4πr² |
| Capacitance in diameter | V = (π/6)d³ |
| surface in diameter | SA = πd² |
🌓 Hemisphere - radius r
| capacity | V = (2/3)πr³ |
| Broadcasting | CSA = 2πr² |
| Total surface area | TSA = 3πr² |
🔺Prism
| capacity | V = Area × Height |
| Broadcasting | LSA = Circumference × Height |
| Total surface area | TSA = LSA + 2 × Subsurface |
V = (½ × feet × height) × length
🔻 Koumbak / Pyramid
| capacity | V = ⅓ × base × height |
| Broadcasting | LSA = ½ × Circumference × Slope Height |
| Total surface area | TSA = LSA + Substrate |
🍩 Ring / Torus
| capacity | V = 2π²Rr² | R = major radius, r = minor radius |
| surface | SA = 4π²Rr |
⚡ Frustum of Cone
r = top radius, R = bottom radius, h = height, l = slope height
| capacity | V = ⅓πh(R² + r² + Rr) |
| Broadcasting | CSA = πl(R + r) |
| Total surface area | TSA = π[R² + r² + l(R + r)] |
| Slope height | l = √[h² + (R − r)²] |
🔄 Key relationships
- Cone : Cylindrical : Sphere(same r, h=2r) → capacitance ratio =1 : 3 : 2
- Cone capacitance= ⅓ × cylindrical capacity (same base, height)
- Hemispherical capacity= ½ × spherical capacitance
- spherical surface= 4 × Area of great circle = 4πr²
📝 Capacity - Solved Examples
10 Important Questions for TNPSC Exam
Question 1: Cube
Question:The perimeter of the cube is 6 cm. If so, find the capacity and total surface area.
Solution:
Margin a = 6 cm.
Capacitance V = a³ = 6³ =216 cubic cm.
Total surface TSA = 6a² = 6 × 36 =216 sq. cm.
Question 2: Cuboid
Question:The length of the tank is 10 m, width 8 m, height 5 m. If so, how many liters of water will it take?
Solution:
l = 10 m., b = 8 m., h = 5 m.
Capacity V = l × b × h = 10 × 8 × 5 =400 cubic m.
1 cubic m. = 1000 litres
∴ Water = 400 × 1000 =4,00,000 liters = 400 kg litres
Question 3: Cylinder
Question:The radius of the cylinder is 7 cm and the height is 10 cm. If so, see capacitance and amplitude. (π = 22/7)
Solution:
Radius r = 7 cm, height h = 10 cm.
Capacitance V = πr²h = (22/7) × 7 × 7 × 10
V = 22 × 7 × 10 =1540 cubic cm.
Area of curvature CSA = 2πrh = 2 × (22/7) × 7 × 10
CSA = 2 × 22 × 10 =440 sq. cm.
Question 4: Cone
Question:The radius of the cone is 6 cm and the height is 8 cm. If so, see capacity and slope height.
Solution:
Radius r = 6 cm, height h = 8 cm.
Height of slope l = √(r² + h²) = √(36 + 64) = √100 =10 cm
Capacitance V = ⅓πr²h = ⅓ × (22/7) × 6 × 6 × 8
V = (22 × 36 × 8) / (7 × 3) = 6336/21 =301.7 cubic cm
Question 5: Sphere
Question:The radius of the sphere is 21 cm. If so, find capacity and surface area.
Solution:
Radius r = 21 cm.
Capacitance V = (4/3)πr³ = (4/3) × (22/7) × 21 × 21 × 21
V = (4 × 22 × 21 × 21 × 21) / (3 × 7)
V = (4 × 22 × 21 × 21 × 3) / 1 =38808 cubic cm.
Area SA = 4πr² = 4 × (22/7) × 21 × 21
SA = 4 × 22 × 63 =5544 sq. cm.
Question 6: Hemisphere
Question:The radius of the hemisphere is 14 cm. If so, find capacity and total surface area.
Solution:
Radius r = 14 cm.
Capacitance V = (2/3)πr³ = (2/3) × (22/7) × 14 × 14 × 14
V = (2 × 22 × 14 × 14 × 2) / 3 = 17248/3 =5749.33 cubic cm.
Total surface area TSA = 3πr² = 3 × (22/7) × 14 × 14
TSA = 3 × 22 × 28 =1848 sq. cm.
Question 7: Melting & Recasting
Question:6 cm Melt the edged cube to 3 cm. How many spheres will you get if you make spheres of radius ?
Solution:
Cubic capacity = a³ = 6³ = 216 cubic cm.
Spherical capacitance = (4/3)πr³ = (4/3) × (22/7) × 27 = (4 × 22 × 27)/(3 × 7)
= 2376/21 = 113.14 cubic cm.
Number of spheres = 216/113.14 ≈1.9 ≈ 1 sphere
(whole spheres only)
Question 8: Cylindrical Tank
Question:2.1 m. The radius is 5 m. A tall cylindrical tank should be painted. If ₹25 per 1 sq m, what is the total cost?
Solution:
r = 2.1 m. = 21/10 m., h = 5 m.
Total surface area TSA = 2πr(r + h)
= 2 × (22/7) × (21/10) × (21/10 + 5)
= 2 × (22/7) × (21/10) × (71/10)
= 2 × 22 × 3 × 71/100 = 93.72 sq m.
Cost = 93.72 × 25 =₹2343
Question 9: Frustum
Question:The top radius of the bucket is 28 cm, the bottom radius is 21 cm, and the height is 24 cm. If so, see capacity.
Solution:
R = 28 cm, r = 21 cm, h = 24 cm.
Capacitance V = ⅓πh(R² + r² + Rr)
= ⅓ × (22/7) × 24 × (784 + 441 + 588)
= ⅓ × (22/7) × 24 × 1813
= (22 × 24 × 1813) / (7 × 3)
= 957528/21 =45596.57 cubic cm. ≈ 45.6 liters
Question 10: Combined Solid
Question:A hemisphere is placed on top of a cylinder. The radius of the cylinder is 7 cm and the height of the cylinder is 10 cm. If so, see total capacity.
Solution:
r = 7 cm, h = 10 cm.
Cylinder capacity = πr²h = (22/7) × 49 × 10 = 1540 cubic cm.
Hemispherical capacitance = (2/3)πr³ = (2/3) × (22/7) × 343
= (2 × 22 × 49) / 3 = 2156/3 = 718.67 cubic cm.
Total capacity = 1540 + 718.67 =2258.67 cubic cm.
📚 Additional practice questions
- The cubic capacity is 512 cubic cm. If, margin =8 cm
- If the cylindrical radius is doubled, the capacitance =4 times
- If radius of sphere is multiplied by 3, capacitance =27 times
- Cone, Cylindrical, Sphere (same r, h=2r) Capacitance ratio =1:3:2
- 1000 liters =1 cubic meter
⚡ Capacitance - Crossroads
Super Tricks to Save Time in TNPSC Exam!
🚀 Trick 1: Key Cylindrical Values
| Radius (r) | Height (h) | Capacitance (πr²h) |
|---|---|---|
| 7 | 1 | 154 |
| 7 | 10 | 1540 |
| 7 | h | 154h |
| 14 | 1 | 616 |
| 21 | 1 | 1386 |
🚀 Trick 2: Dimensional change → Capacitance change
Sphere:Radius n times → Capacity n³ times
Cylinder:r only n times → capacitance n² times
Cylinder:h only n times → capacity n times
| Change | Capacitance change |
|---|---|
| 2 times | 8 times (2³) |
| 3 times | 27 times (3³) |
| ½ times | ⅛ times |
🚀 Trick 3: Cone : Cylindrical : Sphere ratio
Cone : Cylindrical : Sphere =1 : 3 : 2
Example:r = 7, h = 14
Cone = ⅓ × 154 × 14 = 718.67
Cylinder = 154 × 14 = 2156
Sphere = (4/3) × 154 × 7 = 1437.33
Ratio ≈ 1 : 3 : 2 ✓
🚀 Trick 4: Melting account
Old Capacity = New Capacity
Count = Capacity of old / Capacity of a new item
Example:6 cm Make the cube 1 cm. Convert to cubes?
Old capacity = 6³ = 216
New capacity = 1³ = 1
Number = 216/1 =216 cubes
🚀 Trick 5: Liter Conversion Shortcuts
| Change | Quick method |
|---|---|
| Cubic cm → Liter | ÷ 1000 |
| Liter → cubic cm | × 1000 |
| cubic m. → Liter | × 1000 |
| liter → cubic m | ÷ 1000 |
| Cubic cm → ml | Same (1 cm³ = 1 ml) |
🚀 Trick 6: Spherical formula memory
Hemisphere:V = (2/3)πr³ → half of sphere
Spherical Surface:SA = 4πr² → "four by r squared"
| Radius | Capacitance is (4/3)πr³ | The surface area is 4πr² |
|---|---|---|
| 7 | 1437.33 | 616 |
| 14 | 11498.67 | 2464 |
| 21 | 38808 | 5544 |
🚀 Trick 7: Cube quick calculation
= 2 × (first two + middle two + last two)
Example:l=5, b=4, h=3
TSA = 2(20 + 12 + 15) = 2 × 47 =94 square units
🚀 Trick 8: % increase → capacity increase
Increase in capacity = (3x + 3x²/100 + x³/10000)%
Summary (for small x):≈ 3x%
Example:10% increase in sphere radius
Increase in capacitance ≈ 3 × 10 = 30%
Precisely: (1.1)³ = 1.331 → 33.1% increase
🚀 Trick 9: Cone Slope Height Shortcut
3:4:5 ratio: r=3k, h=4k if l=5k
5:12:13 ratio: r=5k, h=12k if l=13k
Example:r=6, h=8
This is a 3:4 ratio (k=2), so l = 5 × 2 =10
🚀 Trick 10: Water filling time
Rate = litres/minute or litres/hour
Example:1000 liter tank, 20 liter/min
Time = 1000/20 =50 minutes
📊 TNPSC Frequently Asked Categories
| Question type | Crossroads |
|---|---|
| cubic capacity | V = a³ |
| Cylinder capacity | V = πr²h (r=7 → 154h) |
| Cone capacitance | V = ⅓πr²h (⅓ of cylinder) |
| Sphere Capacitance | V = (4/3)πr³ |
| Melting and casting | Old V = New V |
| Liter conversion | 1 m³ = 1000 L, 1 cm³ = 1 ml |
💡 Important memory tips
- π = 22/7- If the radius is 7 times
- Cone = ⅓ cylinder(same feet, height)
- Hemisphere = ½ sphere
- 1:3:2- cone:cylindrical:sphere (r equals, h=2r)
- Radius n times → Capacity n³ times
- 1 liter = 1000 cm³ = 1000 ml
- 1 m³ = 1000 liters = 1 kiloliter
- Diagonal of cube = a√3
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