Calculating Volume for Different Shapes

1. Volume of a Cylinder

• Problem: Volume of a cylinder with radius 4 inches and height 6 inches
• Formula: Volume = Base Area × Height
  • Base Area of circle = πr²
  • Volume = πr² × height
• Calculation: R = 4 inches, H = 6 inches
  • π(4)² × 6 = π(16) × 6
  • 16 × 6 = 96
  • Volume = 96π cubic inches
• **Decimal Value: **
  • Use π ≈ 3.14159: 96 × 3.14159 ≈ 301.6 cubic inches
  • Use π ≈ 3.14: 96 × 3.14 ≈ 301.44 cubic inches

2. Volume of a Sphere

• Problem: Surface area of a sphere is 256π square feet. Find its volume.
• Formulas:
  • Surface Area = 4πr²
  • Volume = (4/3)πr³
• Given: Surface Area = 256π
• Calculation to find radius (r):
  • 4πr² = 256π
  • r² = 64
  • r = √64 = 8 feet
• Calculate Volume:
  • Volume = (4/3)π(8)³
  • 8³ = 512
  • (4/3) × 512 = 2048/3
  • Volume = (2048/3)π cubic feet
• Decimal Value:
  • 2048/3 ≈ 682.6667
  • Volume ≈ 682.6667 × 3.14159 ≈ 2144.7 cubic feet

3. Volume of a Cone

• Problem: Volume of a cone with radius 5 cm and height 8 cm
• Formula: Volume = (1/3)πr² × height
• Calculation: R = 5 cm, H = 8 cm
  • π(5)² × 8 / 3
  • 5² = 25
  • 25 × 8 = 200
  • Volume = (200/3)π cubic cm
• Decimal Value:
  • 200/3 ≈ 66.67
  • Volume ≈ 66.67 × 3.14159 ≈ 209.44 cubic cm

4. Volume of a Rectangular Prism

• Problem: Volume of rectangular prism with dimensions 4 ft × 5 ft × 6 ft
• Formula: Volume = Length × Width × Height
• Calculation:
  • 4 × 5 × 6
  • 4 × 5 = 20
  • 20 × 6 = 120
  • Volume = 120 cubic feet

5. Volume of a Cube

• Problem: Longest diagonal of a cube is 12.12436 cm. Find the volume.
• Steps:
  • Use the longest diagonal formula: d = √3 × side length
  • Volume of a cube = side³
• Calculation to find side length (x):
  • 12.12436 = √3 × x
  • x = 12.12436 / √3 ≈ 7 cm
• Calculate Volume:
  • Volume = 7³
  • 7 × 7 × 7 = 343
  • Volume = 343 cubic cm

Conclusion

• Formulas Summary:
  • Cylinder: V = πr²h
  • Sphere: V = (4/3)πr³
  • Cone: V = (1/3)πr²h
  • Rectangular Prism: V = l × w × h
  • Cube: V = side³