Volume Calculator for Common 3D Shapes and Cubic Units
Choose a solid, enter its dimensions and calculate volume with the correct formula. Keep exact fractions and π forms, convert cubic and capacity units, and check every substitution step before using the result.
Last Updated: July 26, 2026
Online Volume Calculator
Use one shared linear unit for every dimension. Integers, decimals, fractions, proper mixed numbers and scientific notation are supported.
Your volume result and formula steps will appear here.
Calculation steps
- Choose a solid and enter its dimensions.
Volume unit conversions
| Volume unit | Equivalent value |
|---|---|
| Calculate a volume with a physical input unit to see conversions. | |
Recent calculations
- Your last six results will appear here.
How to Use This Volume Calculator
- Choose the three-dimensional solid that matches your problem.
- Select one input length unit. Convert every dimension to this unit before entering values.
- Choose the result volume unit and decimal precision. Generic dimensions remain in generic cubic units.
- Enter the requested measurements. Read labels carefully because radius, perpendicular height, semi-axis and straight capsule length have distinct meanings.
- Select Calculate Volume. Review the exact result, decimal value, formula, substitution, steps and conversion table.
Measure the inside dimensions when you need internal capacity. Use outside dimensions when the problem asks for the external solid. Wall thickness, rounded corners and fittings change usable capacity, so a simple geometric result should not be treated as a certified container rating.
What Volume Measures
Volume measures the three-dimensional space occupied by a solid or enclosed inside a container. It differs from area, which measures a flat region, and surface area, which measures a solid’s exterior boundary. A 4 metre by 3 metre floor has an area of 12 m². If a room above it is 2.5 metres high, its rectangular volume is 30 m³.
The unit is cubed because the calculation combines three independent length directions. One cubic metre is a cube one metre long, one metre wide and one metre high. It contains 1,000 litres. One cubic foot is a smaller cube and equals exactly 28.316846592 litres under international length definitions.
Volume Formulas Included
| Solid | Required measurements | Volume formula |
|---|---|---|
| Cube | Side s | V = s³ |
| Rectangular prism | Length l, width w, height h | V = lwh |
| Triangular prism | Triangle base b, triangle height h, perpendicular prism length L | V = bhL/2 |
| Cylinder | Radius r, perpendicular height h | V = πr²h |
| Hollow cylinder | Outer radius R, inner radius r, height h | V = πh(R² − r²) |
| Cone | Radius r, perpendicular height h | V = πr²h/3 |
| Conical frustum | End radii R and r, perpendicular height h | V = πh(R² + Rr + r²)/3 |
| Sphere | Radius r | V = 4πr³/3 |
| Hemisphere | Radius r | V = 2πr³/3 |
| Ellipsoid | Semi-axes a, b and c | V = 4πabc/3 |
| Rectangular pyramid | Base length l, base width w, height h | V = lwh/3 |
| Capsule | Radius r, straight cylinder length c | V = πr²c + 4πr³/3 |
Introductory prism and cylinder reference: OpenStax volume and surface area.
Worked Example: Rectangular Prism Volume
Suppose a rectangular storage space is 12 metres long, 7.5 metres wide and 3 metres high. Multiply all three perpendicular dimensions.
V = 12 × 7.5 × 3
V = 270 m³
The exact answer is 270 m³. Converting the same unrounded value gives 270,000 litres. If one measurement had been entered in centimetres while the selector remained metres, the answer would be wrong. Convert first, then apply the formula.
Worked Example: Cylinder Volume
For a cylinder with radius 3 metres and perpendicular height 10 metres, the circular base area is 9π m². Multiplying by the height gives the volume.
V = π × 3² × 10
V = 90π m³
V ≈ 282.743339 m³
The exact answer remains 90π m³. The decimal is an approximation because π does not terminate. If the problem gives a diameter of 6 metres, divide it by 2 before entering the radius.
Prisms, Pyramids and the One-Third Rule
A prism has the same cross-section throughout its perpendicular length, so its volume equals base area multiplied by prism length. A rectangular prism uses a rectangular base. A triangular prism first calculates the triangular area bh/2, then multiplies it by the distance between the matching triangular ends.
A cone or pyramid tapers to a point. For the same base and perpendicular height, its volume is one third of the corresponding prism or cylinder. The required height is the shortest perpendicular distance from the base plane to the apex. A slanted edge or slant height does not replace it.
Spheres, Ellipsoids and Capsules
Sphere volume uses 4πr³/3, while a hemisphere is exactly half that amount. An ellipsoid stretches a sphere along three perpendicular directions. Enter semi-axes a, b and c, which are half the full axis lengths. When all three semi-axes are equal, the ellipsoid formula reduces to the sphere formula.
A capsule combines a cylinder with two hemispherical ends. Together, the two ends form one sphere. The capsule input c is only the straight cylindrical section between the curved ends. Its full end-to-end length is c + 2r. Setting c to zero creates a sphere, not a zero-volume solid.
Hollow Cylinders and Conical Frustums
Hollow-cylinder mode subtracts the inner cylindrical space from the outer cylinder. The circles are assumed to share the same centre and axis. The outer radius must be at least the inner radius. An inner radius of zero reduces the formula to a solid cylinder. Equal radii or zero height produce exact volume 0.
A conical frustum is the portion left after the tip of a cone is cut by a plane parallel to its base. Its two end radii can be entered in either order because the formula is symmetric. One zero radius gives a cone. Equal positive radii give a cylinder. Both cases are valid boundaries, not errors.
Cubic Unit Conversion
A length factor must be cubed when converting volume. Since 1 metre equals 100 centimetres, 1 m³ equals 100³, or 1,000,000 cm³. Applying the linear factor once, or squaring it as for area, gives the wrong result.
The calculator derives cubic units from exact international length factors. The inch is exactly 0.0254 metre, the foot is exactly 0.3048 metre, the yard is exactly 0.9144 metre and the mile is exactly 1609.344 metres. This gives exactly 1,728 in³ in 1 ft³, 27 ft³ in 1 yd³ and 5,451,776,000 yd³ in 1 mi³.
Conversion reference: NIST revised unit conversion factors.
Litres, Gallons and Acre-Feet
Capacity units can express the same physical volume. One millilitre equals one cubic centimetre. One litre equals 1,000 millilitres or 0.001 m³. A US liquid gallon is exactly 231 cubic inches, equal to 3.785411784 litres. An Imperial gallon is exactly 4.54609 litres, so the two gallon values must not be interchanged.
An international acre-foot equals the volume of one international acre covered to a depth of one international foot. It equals 43,560 ft³ or exactly 1,233.48183754752 m³. The label is explicit because obsolete survey-foot definitions produced slightly different legacy values.
Capacity references: NIST Handbook 44 volume tables and the UK gallon definition.
Zero Dimensions and Degenerate Solids
Zero is accepted as a mathematical boundary. A cube with side zero, a prism with any zero factor, a cylinder with zero radius or height, and a sphere with zero radius all have exact volume 0. The calculator labels these outcomes as degenerate instead of showing 0π or an approximate zero.
Some zero inputs reduce one supported solid to another. A tube with inner radius zero becomes a cylinder. A frustum with one radius zero becomes a cone. A capsule with straight-section length zero becomes a sphere. The result is degenerate only when the complete formula evaluates to zero.
Exact Fractions, π and Rounding
Inputs are parsed into reduced fractions before formula arithmetic. Values such as 1/3, 2 1/2 and 1e-20 do not first pass through ordinary floating-point storage. Prism and pyramid results therefore remain exact rational numbers within the stated limits. Curved solids keep an exact reduced rational coefficient multiplied by π.
The decimal path uses a 120-place π value and converts the unrounded exact coefficient before final display rounding. The selected precision changes presentation only. It does not change the stored fraction, symbolic π result or unit conversion. Small positive values switch to scientific notation instead of being displayed as zero.
Extra displayed digits do not improve the accuracy of real measurements. Match the final significant figures to the quality of your source dimensions and round once at the end.
Important Measurement Assumptions
- Every dimension in one calculation uses the same selected length unit.
- Cylinder, cone, frustum and pyramid height means perpendicular height.
- Triangular-prism height belongs to its triangular base; prism length is the perpendicular distance between the matching triangular ends.
- Hollow-cylinder circles are concentric, and the entered height is shared.
- Ellipsoid values are semi-axes, not full diameters.
- Capsule ends are hemispheres with the same radius as the cylinder.
- Simple formulas ignore wall thickness, rounded corners, seams, fittings and manufacturing tolerances unless their effect is already reflected in the entered dimensions.
Common Volume Mistakes
- Using area or surface area when the problem asks for three-dimensional space.
- Mixing metres, centimetres, feet or inches inside one formula.
- Using a linear or squared conversion factor instead of cubing it.
- Entering diameter where a formula requires radius.
- Using cone or pyramid slant height as perpendicular height.
- Entering full ellipsoid axis lengths instead of semi-axes.
- Treating capsule total length as its straight cylindrical length.
- Assuming US and Imperial gallons are equal.
- Rounding π, fractions or conversions before the final result.
Method, Limits and Result Review
Dimension fields accept up to 50 numeric digits, a maximum raw length of 100 characters and scientific exponents from −100 through 100. Fraction denominators must be nonzero. Proper mixed-number fractional parts must remain smaller than one whole. Negative geometric lengths are rejected.
This page performs direct volume calculations only. It does not calculate surface area, solve missing dimensions, handle partial tank fill, mix feet-and-inches fields, estimate usable capacity allowances, subtract wall thickness, calculate material weight or cost, or model arbitrary composite and irregular solids. Those jobs require different inputs and assumptions.
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Frequently Asked Questions
How do you calculate volume?
Choose the formula for the solid, enter every required length in one shared unit, and evaluate the formula. The result uses the corresponding cubic unit.
Why is volume measured in cubic units?
Volume combines three independent length directions. Multiplying length by width by height therefore produces units such as cm³, m³ or ft³.
Does this volume calculator accept fractions and scientific notation?
Yes. Dimension fields accept integers, decimals, fractions, proper mixed numbers and scientific notation within the stated input limits.
Can I enter dimensions in different units?
No. Convert every length to the selected input unit before calculating. Mixing units in one formula produces an incorrect volume.
How does the calculator handle pi?
Curved-solid results keep an exact reduced coefficient times π. A separate decimal approximation uses a 120-place π value before final rounding.
What happens when a dimension is zero?
Zero is accepted. If it collapses the solid, the calculator returns exact volume 0 and identifies the result as degenerate.
What is the difference between volume and surface area?
Volume measures the three-dimensional space inside a solid. Surface area measures the total area of its outside faces or curved boundary.
How is cylinder volume calculated?
Multiply the circular base area by the perpendicular height: V = πr²h. Radius is half the diameter.
How is cone or pyramid volume calculated?
Multiply the base area by the perpendicular height and divide by 3. A slant height is not the perpendicular height.
How is hollow-cylinder volume calculated?
Subtract the inner cylinder from the outer cylinder: V = πh(R² − r²). The outer radius must be at least the inner radius.
What length does capsule mode use?
Capsule length means the straight cylindrical section between the two hemispherical ends. Total end-to-end length equals that value plus two radii.
Are US and Imperial gallons the same?
No. One US liquid gallon is exactly 3.785411784 litres, while one Imperial gallon is exactly 4.54609 litres. The calculator labels both separately.