Volume
Volume - Solve mathematical problems with step-by-step solutions.
Volume Calculator
For Common 3D Shapes
Formula for a Cylinder
V = πr²h
How the Volume Calculator Works
The Volume Calculator computes the three-dimensional space occupied by various geometric solids including cubes, rectangular prisms, cylinders, spheres, cones, pyramids, and more. Volume is measured in cubic units and is essential in fields ranging from engineering and manufacturing to cooking and everyday problem-solving. Understanding volume calculations helps determine capacity, material requirements, and spatial relationships.
Common Volume Formulas
Cube: V = s3 (side length cubed)
Rectangular Prism (Box): V = l × w × h (length × width × height)
Cylinder: V = πr2h (pi × radius squared × height)
Sphere: V = (4/3)πr3 (4/3 × pi × radius cubed)
Cone: V = (1/3)πr2h (1/3 × pi × radius squared × height)
Pyramid: V = (1/3) × base area × height
Triangular Prism: V = (1/2) × base × height of triangle × length
Ellipsoid: V = (4/3)πabc (where a, b, c are semi-axes)
Understanding Volume
- Cubic Units: Volume is always measured in cubic units (cm3, m3, ft3, etc.)
- Capacity: Volume measures how much space an object occupies or how much it can contain
- Scaling: If you double all dimensions, volume increases by 23 = 8 times
- Density Relationship: Mass = Density × Volume
- Unit Conversions: 1 m3 = 1,000,000 cm3 = 1000 liters; 1 ft3 = 1728 in3 ≈ 7.48 gallons
Volume Calculation Examples
Example 1: Volume of a Cube
Given: A cube with side length s = 5 cm
Formula: V = s3
Calculation: V = 53 = 125 cm3
Answer: The volume is 125 cubic centimeters
Example 2: Volume of a Rectangular Prism (Box)
Given: A box with length 10 in, width 6 in, height 4 in
Formula: V = l × w × h
Calculation: V = 10 × 6 × 4 = 240 in3
Answer: The volume is 240 cubic inches
Example 3: Volume of a Cylinder
Given: A cylinder with radius r = 3 m and height h = 8 m
Formula: V = πr2h
Calculation: V = π × (3)2 × 8 = π × 9 × 8 = 72π ≈ 226.19 m3
Answer: The volume is approximately 226.19 cubic meters
Example 4: Volume of a Sphere
Given: A sphere with radius r = 6 feet
Formula: V = (4/3)πr3
Calculation: V = (4/3) × π × (6)3 = (4/3) × π × 216 = 288π ≈ 904.78 ft3
Answer: The volume is approximately 904.78 cubic feet
Example 5: Volume of a Cone
Given: A cone with base radius r = 4 cm and height h = 9 cm
Formula: V = (1/3)πr2h
Calculation: V = (1/3) × π × (4)2 × 9 = (1/3) × π × 16 × 9 = 48π ≈ 150.80 cm3
Answer: The volume is approximately 150.80 cubic centimeters
Example 6: Volume of a Square Pyramid
Given: Square base with side 6 m, height 10 m
Base Area: A = 62 = 36 m2
Formula: V = (1/3) × base area × height
Calculation: V = (1/3) × 36 × 10 = 120 m3
Answer: The volume is 120 cubic meters
Example 7: Volume of Water in a Pool
Problem: Rectangular pool 25 m long, 10 m wide, 2 m deep. How many liters?
Volume: V = 25 × 10 × 2 = 500 m3
Convert to Liters: 500 m3 × 1000 L/m3 = 500,000 liters
Answer: The pool holds 500,000 liters (500 kiloliters) of water
Tips for Calculating Volume
Essential Volume Calculation Tips
- Cubic Units: Always express volume in cubic units (cm3, m3, ft3). Don't confuse with area (square units)
- Unit Consistency: All dimensions must be in the same units before calculating
- Radius vs Diameter: Many formulas use radius. If given diameter, divide by 2 first
- The 1/3 Factor: Cones and pyramids have 1/3 in their formulas—they're 1/3 the volume of corresponding cylinders/prisms
- Pi Precision: Use π = 3.14159 or your calculator's π button for accuracy
- Compound Shapes: Break complex objects into simpler shapes, calculate each volume, then add or subtract
- Volume Scaling: Doubling linear dimensions multiplies volume by 8 (23); tripling multiplies by 27 (33)
- Liquid Conversions: Remember 1 m3 = 1000 liters, 1 ft3 ≈ 7.48 gallons, 1 gallon ≈ 3.785 liters
- Hollow Objects: Subtract inner volume from outer volume to find material volume
- Practical Tip: For irregular objects, use water displacement method (Archimedes' principle)
Volume Unit Conversions
Volume Unit Conversions
Metric System:
- 1 m3 = 1,000,000 cm3 = 1,000,000,000 mm3
- 1 m3 = 1,000 liters (L) = 1,000,000 milliliters (mL)
- 1 liter = 1,000 cm3 = 1 dm3
- 1 cm3 = 1 mL (milliliter)
Imperial/US System:
- 1 ft3 = 1,728 in3 = 7.481 gallons (US)
- 1 yd3 = 27 ft3 = 46,656 in3
- 1 gallon (US) = 231 in3 ≈ 3.785 liters
- 1 gallon (UK) ≈ 4.546 liters
Metric to Imperial:
- 1 m3 ≈ 35.315 ft3 ≈ 264.17 gallons (US)
- 1 liter ≈ 0.264 gallons (US) ≈ 0.220 gallons (UK)
- 1 ft3 ≈ 0.0283 m3 ≈ 28.317 liters
Conversion Tip:
When converting volume units, cube the linear conversion factor. For example, since 1 m = 100 cm, then 1 m3 = (100)3 cm3 = 1,000,000 cm3
Real-World Applications
Real-World Applications
- Construction: Calculating concrete needed for foundations, walls, and columns
- Manufacturing: Determining material requirements, packaging sizes, and container capacity
- Shipping and Logistics: Computing cargo space, freight charges, and storage capacity
- Engineering: Fluid dynamics, pressure calculations, and structural analysis
- Medicine: Dosage calculations, organ volumes, and medical imaging
- Cooking and Baking: Recipe conversions, container selection, and ingredient measurements
- Agriculture: Grain silo capacity, irrigation volumes, and fertilizer quantities
- Environmental Science: Water reservoir volumes, waste management, and pollution dispersion
- Architecture: Room volumes for HVAC design, acoustic calculations, and space planning
- Automotive: Engine displacement, fuel tank capacity, and cargo space
Frequently Asked Questions
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