Volume Calculator
Select a shape → choose units → enter dimensions → get volume in m³, cm³, ft³, litres and gallons instantly
Volume in All Units
What Is Volume? The Complete Guide
Definitions, units, the difference between volume and capacity, and why volume matters in science and everyday life
Volume is the measure of the three-dimensional space enclosed by or occupied by an object. It answers the question: how much space does this thing take up? Volume is fundamental to physics, chemistry, engineering, architecture, medicine, cooking and everyday life. It is always expressed in cubic units — cubic metres (m³), cubic centimetres (cm³), cubic feet (ft³) — because it is the product of three linear dimensions.
The concept of volume dates back to ancient times. Archimedes famously determined the volume of an irregular object by measuring water displacement — still a valid method today for objects too complex for formulae. For regular geometric solids, precise mathematical formulas exist for every shape, derived from calculus and geometry.
The relationship between volume and capacity is subtle but important. Volume measures the 3D space an object occupies (solid or hollow). Capacity refers specifically to how much liquid or gas a hollow container can hold. For practical purposes they are equal for thin-walled containers. The SI unit of capacity is the litre, where 1 litre = 1 cubic decimetre (dm³) = 1,000 cm³ = 0.001 m³.
Volume Formulas for All 12 Shapes — Complete Reference
Every formula with variables defined, worked examples and real-world applications
| Shape | Formula | Variables | Worked Example | Real-World Use |
|---|---|---|---|---|
| 🧊 Cube | V = a³ | a = side length | a=4 cm → 64 cm³ | Dice, sugar cubes, storage boxes |
| 📦 Rectangular Prism | V = l × w × h | l=length, w=width, h=height | 4×3×2 m → 24 m³ | Rooms, swimming pools, tanks, packaging |
| 🔵 Sphere | V = (4/3)πr³ | r = radius | r=5 cm → 523.6 cm³ | Balls, bubbles, planets, balloons |
| 🌏 Hemisphere | V = (2/3)πr³ | r = radius | r=3 m → 56.55 m³ | Domes, bowls, half-sphere tanks |
| 🥫 Cylinder | V = πr²h | r=base radius, h=height | r=2, h=5 m → 62.83 m³ | Pipes, tanks, cans, columns |
| 🍦 Cone | V = (1/3)πr²h | r=base radius, h=height | r=3, h=6 cm → 56.55 cm³ | Funnels, ice cream cones, conical silos |
| 🔺 Square Pyramid | V = (1/3)a²h | a=base side, h=height | a=6, h=9 m → 108 m³ | Egyptian pyramids, rooftops, hoppers |
| 📐 Triangular Prism | V = (1/2)b·t·l | b=base, t=tri. height, l=length | b=4, t=3, l=8 m → 48 m³ | Roofs, wedge ramps, Toblerone |
| 🥚 Ellipsoid | V = (4/3)πabc | a,b,c = 3 semi-axes | a=3,b=2,c=1 m → 25.13 m³ | Eggs, rugby balls, tanks, planets |
| 💊 Capsule | V = πr²(h + 4r/3) | r=radius, h=cylinder height | r=1, h=3 cm → 13.61 cm³ | Medicine capsules, tanks, submarines |
| 🍩 Torus | V = 2π²Rr² | R=major radius, r=minor radius | R=4, r=1 cm → 78.96 cm³ | Donuts, rings, O-rings, tire tubes |
| 🪣 Frustum | V = (πh/3)(R²+Rr+r²) | R=bottom radius, r=top, h=height | R=4,r=2,h=6 m → 263 m³ | Buckets, flower pots, cooling towers |
Math.PI = 3.141592653589793 for all calculations.Visual Formula Cards — Shape by Shape
Each shape's formula with worked example, variable definitions and practical applications
Example: a = 5 cm → V = 5³ = 125 cm³
A cube is a special rectangular prism where l = w = h. Doubling a side multiplies volume by 8.
Example: 4m × 3m × 2m = 24 m³
Most common shape in construction and packaging. Also called a cuboid or box.
Example: r = 5 cm → (4/3)π(125) = 523.6 cm³
A sphere has the smallest surface area for a given volume — nature's most efficient shape.
Example: r = 4 m → (2/3)π(64) = 134.04 m³
Exactly half a sphere's volume. Used for domes, bowls and half-sphere storage tanks.
Example: r=3, h=8 cm → π×9×8 = 226.2 cm³
The most common shape for containers (cans, tanks, pipes). Prism with circular cross-section.
Example: r=3, h=9 cm → (1/3)π×9×9 = 84.82 cm³
Exactly ⅓ the volume of a cylinder with the same base and height.
Example: a=6m, h=9m → (1/3)×36×9 = 108 m³
Great Pyramid of Giza: base 230.4m, height 138.8m → V ≈ 2,459,712 m³!
Example: b=4, t=3, l=10 m → ½×4×3×10 = 60 m³
Used for ramp volumes, roof structures, Toblerone-shaped packages.
Example: a=3,b=2,c=1.5 m → (4/3)π×9 = 37.70 m³
A sphere is a special ellipsoid where a=b=c=r. Earth is an oblate spheroid (a=b>c).
Example: r=1cm, h=4cm → π×1×(4+1.33) = 16.76 cm³
Cylinder capped by two hemispheres. Used for pharmaceutical capsules and submarine hulls.
Example: R=5, r=1.5 cm → 2π²×5×2.25 = 222.1 cm³
Donut shape. Used for O-rings, life preservers, architectural ring structures.
Example: R=4,r=2,h=6 m → (π×6/3)×(16+8+4) = 175.9 m³
Buckets, conical hoppers, flower pots, cooling towers, wine glasses.
How to Use the Volume Calculator
Step-by-step guide for students, engineers, architects and everyday users
- 1
Select Your Shape
Click one of the 12 shape tiles. Each shape is illustrated with an emoji and labelled clearly. The input fields update automatically to show only the dimensions relevant to the selected shape — for example, selecting Cylinder shows radius and height fields; selecting Rectangular Prism shows length, width and height.
- 2
Choose Your Input Unit
Select the unit your measurements are in: metres (m), centimetres (cm), millimetres (mm), feet (ft), inches (in) or yards (yd). You can mix shapes but must keep all dimensions in the same unit within one calculation. All output appears simultaneously in 6 different units regardless of input unit chosen.
- 3
Enter the Dimensions
Type the measurements into the input fields. For a Cube: just one side length. For a Sphere: radius only. For a Cylinder: radius and height. For a Rectangular Prism: length, width and height. For complex shapes like Ellipsoid (3 semi-axes) or Torus (major and minor radius), more fields appear. Decimal values are fully supported.
- 4
Click Calculate Volume
Press the green "Calculate Volume" button. The result hero panel shows the primary volume in cubic metres (m³). The "Volume in All Units" grid simultaneously displays the result in cm³, ft³, in³, litres and US gallons — saving you from manual unit conversions.
- 5
Review the Step-by-Step Working
Click "Step-by-Step Working" to expand the full solution panel. It shows the formula name, variable substitutions with your actual values, intermediate calculations and the final result. This is particularly useful for students checking homework, teachers preparing examples, or engineers validating calculations.
Volume in Science, Nature & Engineering — Key Facts
Fascinating applications of volume across physics, chemistry, biology and engineering
Volume of the Earth
Earth is an oblate spheroid with an equatorial radius of 6,371 km. Using the sphere formula, V ≈ (4/3)π×(6.371×10⁶)³ ≈ 1.083 × 10²¹ m³ (1.083 trillion km³). The volume of all Earth's oceans combined is about 1.335 billion km³ — just 0.12% of Earth's total volume.
The Sphere: Nature's Most Efficient Shape
Among all shapes with the same surface area, the sphere has the greatest volume (isoperimetric inequality). This is why bubbles, water droplets, and cells are spherical — nature minimises surface energy. A sphere with surface area 100 cm² has 29.6% more volume than a cube with the same surface area.
The Great Pyramid's Volume
The Great Pyramid of Giza has a square base of 230.4 m and original height of 146.5 m. Volume = (1/3) × 230.4² × 146.5 = 2,592,276 m³ — enough to fill the Empire State Building almost twice. This is approximately 2.3 million cubic metres of limestone, granite and mortar.
Volume in Biochemistry
Human red blood cells are biconcave discs with a volume of approximately 90 fL (femtolitres = 10⁻¹⁵ L). An adult human body contains about 5 litres of blood with ~25 trillion red blood cells. The human cell nucleus has an average volume of ~500 µm³. DNA in a single nucleus, if stretched out, would be ~2 metres long — all packed into a 6 µm sphere.
Volume Scaling in Engineering
When you scale a structure up by factor k, its volume (and mass) scales by k³ but its cross-sectional area (structural strength) scales by only k². This is why large animals need proportionally thicker bones than small animals (the square-cube law), and why you can't simply scale up a model bridge to full size without redesigning the structure.
Practical Volume: Containers
Standard volume benchmarks: 1 teaspoon = 5 mL. 1 cup = 250 mL. 1 litre bottle = 1,000 cm³ (10×10×10 cm cube). Standard bathtub ≈ 200–300 litres. Swimming pool (Olympic) = 2,500 m³ = 2.5 million litres. Fuel tanker ≈ 30,000 litres. Supertanker ≈ 500,000 m³.
Volume and Temperature (Gases)
For ideal gases, volume is directly proportional to temperature (at constant pressure): V ∝ T (Charles's Law). A balloon that occupies 10 L at 20°C (293K) will shrink to 8.3 L when cooled to −20°C (253K). This is why car tyres appear deflated in cold weather and why hot air balloons rise — the heated air inside is less dense (larger volume per mole of gas).
Volume of Irregular Shapes
For complex irregular shapes, volume is calculated by: (1) Water displacement (Archimedes' method), (2) Numerical integration (breaking the shape into thin slices — used in CAD), (3) 3D scanning + mesh analysis (modern industrial method), or (4) Monte Carlo simulation (random point sampling — used in physics and finance). All ultimately reduce to summing infinitesimally small regular volumes.
Volume Unit Conversions — Complete Reference
Every volume unit conversion factor you need — from millilitres to cubic miles
Volume units span an enormous range — from the femtolitre (10⁻¹⁵ L) used in cell biology to the cubic kilometre (10⁹ m³) used in oceanography. Understanding the conversion factors between the most common units is essential for science, engineering and everyday calculations.
| Unit | Symbol | Equivalent in m³ | Equivalent in Litres | Commonly Used For |
|---|---|---|---|---|
| Cubic millimetre | mm³ | 10⁻⁹ m³ | 0.000001 L | Engineering tolerances, microfluidics |
| Millilitre | mL | 10⁻⁶ m³ | 0.001 L | Medicine, cooking, lab measurements |
| Cubic centimetre | cm³ | 10⁻⁶ m³ | 0.001 L | Same as millilitre — chemistry, medicine |
| Teaspoon (metric) | tsp | 5 × 10⁻⁶ m³ | 0.005 L | Cooking and recipes |
| Tablespoon (metric) | tbsp | 15 × 10⁻⁶ m³ | 0.015 L | Cooking and recipes |
| Fluid ounce (US) | fl oz | 2.957 × 10⁻⁵ m³ | 0.02957 L | Beverages (USA) |
| Cup (US) | cup | 2.366 × 10⁻⁴ m³ | 0.2366 L | Cooking (USA) |
| Pint (US liquid) | pt | 4.731 × 10⁻⁴ m³ | 0.4731 L | Beverages (USA) |
| Litre | L | 0.001 m³ | 1 L | Beverages, fuel, everyday volumes |
| Quart (US) | qt | 9.464 × 10⁻⁴ m³ | 0.9464 L | Beverages, paint (USA) |
| Gallon (US) | gal | 0.003785 m³ | 3.785 L | Fuel, paint, large liquids (USA) |
| Cubic foot | ft³ | 0.02832 m³ | 28.32 L | Concrete, lumber, gas (USA/UK) |
| Cubic yard | yd³ | 0.7646 m³ | 764.6 L | Concrete, soil, aggregates (USA) |
| Cubic metre | m³ | 1 m³ | 1,000 L | Construction, engineering, SI standard |
| Cubic kilometre | km³ | 10⁹ m³ | 10¹² L | Oceans, glaciers, reservoirs |
Frequently Asked Questions
Common questions about volume formulas, unit conversions and real-world applications