Calculator
Engineering
MathVolume Calculator
Volume calculator for sphere, cone, cube, cylinder, tank, capsule, pyramid and ellipsoid. Converts m³/L/gal/ft³ and shows fill weight by density.
The Volume Calculator helps you calculate the volume of common three-dimensional shapes. Select a shape, enter dimensions, and get instant volume calculations with formulas and explanations.
Sphere
Cone
Cube
Cylinder
Rectangular Tank
Square Pyramid
Tube (Hollow Cylinder)
Capsule
Conical Frustum
Ellipsoid
Have feedback? Report bugs, suggest features, or share your thoughts — we read them allWhat is Volume?
Volume is the measure of three-dimensional space occupied by an object or enclosed within a container. It's expressed in cubic units (m³, ft³, cm³) or liquid measures (liters, gallons). Volume calculations are essential for tanks, containers, material quantities, concrete pours, liquid storage, and capacity planning. Each 3D shape has a specific formula based on its geometry.
Volume Formulas by Shape
Sphere: V = (4/3) × π × r³
Cone: V = (1/3) × π × r² × h
Cube: V = side³
Cylinder: V = π × r² × h
Rectangular Tank: V = L × W × H
Capsule: V = π × r² × h + (4/3) × π × r³
Conical Frustum: V = (π×h/3) × (r₁² + r₁×r₂ + r₂²)
Ellipsoid: V = (4/3) × π × a × b × c
Square Pyramid: V = (1/3) × base² × h
Tube: V = π × (R² - r²) × h
Applications
- Construction: Concrete volumes, material quantities
- Storage: Tank capacity, container sizing
- Manufacturing: Material volume, packaging
- Agriculture: Silo capacity, grain storage
- Water management: Reservoir capacity, pipe volume
- Chemistry: Reactor sizing, batch volumes
- Shipping: Cargo volume, freight calculations
Tips for Volume Calculations
- Always use consistent units throughout calculation
- 1 cubic meter (m³) = 1000 liters = 264.17 gallons
- 1 cubic foot (ft³) = 7.48 gallons = 28.32 liters
- For tanks, account for wall thickness in capacity calculations
- Consider fill level - rarely filled to 100% capacity
- Add volume for fittings, pipes, and dead space in systems
- Weight = Volume × Density (water: 1000 kg/m³)
Frequently Asked Questions
Fill weight is simply mass = volume × density. This calculator does it for you: compute the volume, then pick a substance (or enter a custom kg/m³) and it shows the contained mass in kilograms, tonnes and pounds. The densities used are Water 1000, Seawater 1025, Diesel 832, Gasoline 745, Crude oil 870, Milk 1030, Concrete 2400 and dry Sand 1600 kg/m³. Example: a 9 m³ water tank holds 9 m³ × 1000 kg/m³ = 9,000 kg = 9 tonnes — a real structural load case, not a curiosity, and the reason engineers size foundations, racking, pump lift and freight from weight rather than volume. For partly filled vessels, multiply by the actual fill fraction. For materials that settle or compact (sand, grain, aggregate), use bulk density (loose vs compacted differs 10–25%) rather than solid density.
Enter the tank dimensions, choose your unit (m, ft, cm, in) and read the L / US gal / ft³ row alongside the m³ result. The key conversions: 1 m³ = 1,000 liters = 264.17 US gallons = 35.31 ft³; 1 ft³ = 7.481 US gallons = 28.32 liters; 1 US gallon = 3.7854 L (the UK/Imperial gallon is 4.5461 L, about 20% larger). For a 2 m × 3 m × 1.5 m rectangular tank: 9 m³ = 9,000 L = 2,378 US gallons. Remember that geometric volume is the theoretical maximum — usable (working) capacity is typically 5–15% less because of overflow space, expansion room, inlet/outlet geometry and internal fittings, so always derate when sizing real storage.
Three of the most-used 3D shapes: Cylinder V = π × r² × h (a stack of circles); Sphere V = (4/3) × π × r³ (Archimedes' classical result, derivable by Cavalieri's principle from a hemisphere inscribed in a cylinder); Cone V = (1/3) × π × r² × h (exactly one-third the cylinder of the same base and height — a result that holds for any pyramid). Example: a 1 m radius, 2 m tall cylinder holds π × 1 × 2 = 6.283 m³ (6,283 liters); a 1 m radius sphere holds 4.189 m³; a 1 m radius, 2 m tall cone holds 2.094 m³. The cone-cylinder factor of 1/3 was proved by Eudoxus in the 4th century BCE — Archimedes was so proud of his sphere-volume derivation that he asked for it engraved on his tombstone.
This result, known to Eudoxus and Archimedes, follows from Cavalieri's principle (any two solids with the same cross-sectional area at every height have equal volume) plus the fact that a cube can be partitioned into three congruent square pyramids. More rigorously, slice the cone into thin horizontal disks at height y; each has radius r(y) = R × (1 − y/h) and area π R² (1 − y/h)². Integrating from 0 to h: V = π R² × ∫₀ʰ (1 − y/h)² dy = π R² × h × [−(1 − y/h)³/3]₀ʰ = π R² h / 3. The same 1/3 factor applies to any pyramid (square, triangular, hexagonal base), making it a universal property of "shapes that taper linearly to a point."
Rectangular box volume is V = L × W × H — simply multiply length × width × height in consistent units. For a 2 m × 3 m × 1.5 m water tank: V = 9 m³ = 9,000 liters = 2,378 US gallons. For irregular shapes that can be approximated as rectangular sections (additive decomposition), compute each section's volume and sum. Beware of internal obstructions — heaters, baffles, instruments — which can reduce usable volume by 5–15% in process tanks. For sloped floors common in liquid-storage tanks, use the prismatoid formula V = (h/6)(A_top + 4A_mid + A_bottom), Simpson's rule applied to area. Always specify whether the volume is geometric (theoretical) or net working capacity (between overflow and low-level cutout).
A capsule is the shape of many pressure vessels, propane tanks, and LPG storage: a cylindrical body of length L and radius r, capped by two hemispheres at each end. Total volume V = π r² L + (4/3) π r³ = π r² (L + 4r/3). For a propane tank with r = 0.3 m and cylindrical length L = 1.2 m: V = π × 0.09 × (1.2 + 0.4) = 0.452 m³ (452 liters). For partial fill (most fuel tanks are not full), the partial volume requires the spherical-cap formula plus circular-segment formula and is best computed numerically. Capsules are popular because hemispherical heads handle pressure efficiently (uniform hoop stress) compared with flat ends, which require thick plate and reinforcement per ASME Section VIII Division 1.
The prismatoid formula V = (h/6)(A₁ + 4A_m + A₂) computes the volume of any solid bounded by two parallel planes at distance h, where A₁ is the bottom area, A₂ is the top area, and A_m is the area at the midplane. It is exact for prisms, cones, pyramids, spheres (vertical slice), frusta, and any shape whose cross-section is a polynomial of degree ≤ 3 in the height variable — a remarkable generalization. For a frustum of a cone (cylinder tapering linearly between two radii r₁ and r₂), it simplifies to V = (π h/3)(r₁² + r₁ r₂ + r₂²). Civil engineers use the prismatoid formula extensively for earthwork volume between station cross-sections in road, dam, and excavation projects.
Three practical methods: (1) Water displacement (Archimedes principle) — submerge the object in a graduated container and measure volume of water displaced; precise to a few mL for hand-held objects, used for organ volume in biology and casting volumes in foundry work. (2) 3D scanning — laser or structured-light scanners produce a mesh, then software like MeshLab or Geomagic computes enclosed volume via divergence theorem. (3) Numerical integration on a known cross-section function: slice the object into thin layers, measure each area (photo + image processing, or CAD), and apply Simpson's rule: V ≈ (h/3) × (A₀ + 4A₁ + 2A₂ + 4A₃ + … + Aₙ). Hydrostatic weighing (mass in air vs mass in water) is another classic method, accurate to 0.01% for samples up to a few kg.
Three closely related but distinct concepts: Volume is the geometric measure of space (m³, L, ft³, gal). Capacity refers to the usable volume a container can hold under specified conditions — a 100 L fuel tank may have 95 L working capacity because of expansion space and inlet/outlet geometry. Displacement is the volume of fluid pushed aside by an immersed solid (ships, pistons): an engine "2.0 L displacement" means the pistons sweep 2.0 L per cycle, not the engine's external volume. Naval architects also distinguish gross tonnage (a volumetric measure of enclosed space, 1 GT ≈ 2.83 m³) from deadweight tonnage (mass-carrying capacity). When ordering tanks, vessels, or vehicles, always check which definition the manufacturer uses — confusion has led to expensive sizing errors in fuel storage, ballast, and pressure vessel projects.
Memorize the key relationships: 1 m³ = 1,000 liters = 1,000,000 cm³ = 1,000,000 mL; 1 US gallon = 3.7854 L = 231 in³; 1 Imperial (UK) gallon = 4.5461 L (20% larger than US); 1 ft³ = 28.317 L = 7.481 US gal; 1 barrel (oil, US) = 159 L = 42 US gal; 1 acre-foot (irrigation) = 1,233.5 m³. The US/UK gallon difference still trips up engineers translating between American and British references — a Brit and an American both saying "50 gallons of fuel" actually mean different amounts. For chemistry, density bridges mass and volume: 1 L of water at 4°C weighs exactly 1.000 kg (by historical definition before redefinition of the kilogram); other fluids vary (diesel 0.832 kg/L, mercury 13.534 kg/L). Always specify which gallon, and prefer SI units in technical documents.
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