Volume Formulas: Every Common 3D Shape
Volume measures the three-dimensional space inside a solid object, expressed in cubic units (cm³, m³, ft³, litres). Each shape has a specific formula derived from its geometry.
Core Volume Formulas
- Cube: V = s³
- Rectangular box (cuboid): V = l × w × h
- Cylinder: V = π × r² × h
- Sphere: V = (4/3) × π × r³
- Cone: V = (1/3) × π × r² × h
- Pyramid: V = (1/3) × base area × h
- Ellipsoid: V = (4/3) × π × a × b × c
Worked Examples
- Cylinder r=5cm, h=10cm: V = π×25×10 = 785.4 cm³
- Sphere r=3cm: V = (4/3)×π×27 = 113.1 cm³
- Cone r=4cm, h=9cm: V = (1/3)×π×16×9 = 150.8 cm³
Volume Unit Conversions
- 1 litre = 1,000 cm³ = 1 dm³
- 1 m³ = 1,000 litres
- 1 US gallon = 3,785.4 cm³
- 1 ft³ = 28.317 litres
Calculate volume for any shape: Free Volume Calculator
Volume Formulas for Common Shapes
- Cube: V = a³ (a = side length)
- Rectangular box: V = l × w × h
- Cylinder: V = π × r² × h
- Sphere: V = (4/3) × π × r³
- Cone: V = (1/3) × π × r² × h
- Triangular prism: V = (1/2) × b × h × l
- Pyramid: V = (1/3) × base area × height
Real-World Applications
Volume calculations underpin countless everyday and engineering tasks. Concrete orders for a foundation slab (rectangular volume), paint coverage for a room (volume of walls approximated as rectangular prisms), pool capacity in litres (cylindrical or rectangular), dosage of liquid medications (volume in mL), fuel tank capacity (often cylindrical or irregular shapes approximated), and shipping container sizing all require volume calculations. In cooking, understanding that a recipe's yield depends on volume helps scale portions correctly. Environmental engineers calculate reservoir volumes to plan water supply and irrigation systems.
Frequently Asked Questions
How many litres are in a cubic metre?
Exactly 1,000 litres. Since 1 litre = 1 decimetre³ = 0.001 m³, it follows that 1 m³ = 1,000 L. A cubic metre of water weighs exactly 1,000 kg (1 tonne) at 4°C — the temperature at which water is densest.
How do I calculate the volume of an irregular shape?
For irregular objects, use water displacement: submerge the object in a measured container of water and measure the rise in water level. Volume of object = volume of water displaced. Archimedes discovered this principle over 2,000 years ago. For irregular solid shapes in CAD, software integrates cross-sectional areas automatically.
What is the volume of a standard wine bottle?
A standard wine bottle holds 750 mL = 0.75 L = 0.00075 m³. A magnum is 1.5 L (double). A Jeroboam is 3 L (four bottles). The cylindrical shape has an internal diameter of about 7.5 cm and height of about 28 cm: π × 3.75² × 28 ÷ 1000 ≈ 1.24 L, though the bottle neck, punt, and glass thickness reduce the internal volume to 750 mL.