Volume Calculator
Compute the volume of 15 common solids. Pick a shape, enter dimensions, and get an instant result with a live labeled diagram, cubic-unit conversions (m³, ft³, yd³, liters, US & imperial gallons), cost and material weight estimators, and full step-by-step working.
Composite shape builder
Build a composite object out of any number of the 13 solids above and get the combined total volume with a per-component breakdown. Useful for tanks with domed heads, buildings with roofs, prosthetics, complex castings, or any shape you can decompose into simpler solids.
Partial fill level
Find the volume and percent full of a horizontal cylindrical or spherical tank given the liquid height from the bottom.
What is volume?
Volume is the amount of three-dimensional space a solid object fills. If you could pour water into the shape until it was completely full, the amount of water inside would be the volume. It answers questions like “how much fits inside?”, “how much material do I need to make this?” and “how much space does this take up on a shelf, in a truck or in a tank?”.
Volume is always measured in cubic units — cubic centimeters, cubic meters, cubic feet, cubic inches — or in liquid units such as liters and gallons. That is because volume extends in three directions at the same time: length, width and height. A single number like “50” means nothing on its own; 50 cm³ and 50 m³ are different by a factor of one million.
Every volume formula on this page is built from one simple idea: take a cross-section, measure how much area it covers, then multiply by how far the shape stretches in the third direction. When the shape narrows to a point (a cone or pyramid) you divide by three. When it curves (a sphere, cylinder or torus) you add a π. Once you notice that pattern, the formulas stop feeling like a list to memorise and start feeling like small variations of the same rule.
Volume, shape by shape
One card per solid — the geometric idea, its formula with each variable spelled out, the same diagram you see above the results panel, and one hand-checked numeric example.
Sphere
A sphere is every point the same distance r from a centre — a perfect ball. Its volume comes from stacking infinitely many circular cross-sections and simplifies to a clean cubic in r.
Cone
A cone tapers from a circular base to a single apex. The (1/3) factor is the same one that appears in every pyramid — a tapered solid holds exactly one-third of the prism that surrounds it.
Cube
A cube is six equal square faces meeting at right angles. Because every edge has the same length s, volume is just that length used three times over.
Cylinder
A cylinder is a circle stretched straight upward. The base area π r² multiplied by the height h gives the volume — the same base × height pattern every prism follows.
Rectangular tank (box)
A rectangular tank — the shape of most rooms, boxes and pools — has three independent edges. Volume is simply the product of the three side lengths.
Capsule
A capsule is a cylinder with a hemisphere glued onto each end. Adding the two hemispheres together makes a full sphere, so its volume is just cylinder + sphere.
Spherical cap
A spherical cap is the slice you get when a flat plane cuts a ball. You only need any two of ball radius R, cap height h, or base radius a — the third is fixed by a² = h(2R − h).
Conical frustum
A frustum is what's left when you slice the tip off a cone with a cut parallel to the base — a bucket shape. The r₁² + r₁r₂ + r₂² term averages the two circular ends together.
Ellipsoid
An ellipsoid is a stretched or squashed sphere with three independent semi-axes a, b and c. Setting all three equal recovers the sphere formula (4/3) π r³.
Square / rectangular pyramid
A pyramid tapers from a rectangular base to a single apex. Like the cone, a tapered solid holds one-third of the prism that would enclose it — hence the 1/3.
Tube (hollow cylinder)
A tube is a cylinder with a smaller cylinder removed from its centre — a pipe. Subtract the inner circle's area from the outer circle's, then multiply by the length.
Triangular prism
A triangular prism is a triangle extruded along a straight length. Volume is the triangle's area A times that length L — enter the triangle by base + height, or by three sides (Heron's formula).
Torus
A torus is a donut — a small circle of radius r swept around a larger central circle of radius R. The formula falls out of Pappus's theorem: tube area π r² times the sweep circumference 2π R.
Tetrahedron
A regular tetrahedron is the simplest Platonic solid — four equilateral-triangle faces meeting at four vertices. Every edge has the same length s, so both volume and surface area reduce to a single-variable formula.
Octahedron
A regular octahedron is eight equilateral-triangle faces arranged like two square pyramids glued base to base. Like the tetrahedron, a single edge length s determines the whole solid.
Common volume unit conversions
| From | To | Multiply by |
|---|---|---|
| 1 cubic meter (m³) | cubic feet | 35.3147 |
| 1 cubic meter (m³) | cubic yards | 1.30795 |
| 1 cubic meter (m³) | liters | 1000 |
| 1 cubic meter (m³) | US gallons | 264.1720 |
| 1 cubic meter (m³) | imperial gallons | 219.9692 |
| 1 cubic foot (ft³) | cubic meters | 0.028317 |
| 1 cubic foot (ft³) | liters | 28.3168 |
| 1 cubic foot (ft³) | US gallons | 7.48052 |
| 1 cubic yard (yd³) | cubic meters | 0.764555 |
| 1 liter (L) | cubic meters | 0.001 |
| 1 liter (L) | US gallons | 0.264172 |
| 1 US gallon | liters | 3.78541 |
| 1 imperial gallon | liters | 4.54609 |
| 1 imperial gallon | US gallons | 1.20095 |
Features of this calculator
- 15 solids — sphere, cone, cube, cylinder, rectangular tank, capsule, spherical cap, conical frustum, ellipsoid, pyramid (square or rectangular base), tube (hollow cylinder), triangular prism, torus, tetrahedron, and octahedron.
- Live SVG diagram for every shape that updates as you type, with each entered dimension labeled directly on the figure.
- Spherical cap accepts any two of ball radius R, base radius a, and cap height h — the calculator derives the third from a² = h(2R − h).
- Triangular prism supports both base–height and three-sides (Heron's formula) triangle input — a shape most volume calculators skip.
- Torus and tube (hollow cylinder) built in — two more gaps that most competitors miss.
- Length unit selector (mm, cm, m, km, in, ft, yd); every result is instantly re-shown in cubic meters, cubic feet, cubic yards, cubic inches, liters, US gallons and imperial gallons.
- Cost / material estimator on every shape — enter a price per cubic unit and get an instant total for concrete, soil, water, gravel, mulch, resin or fill.
- Full step-by-step working — formula, substitution and intermediate derivations shown for every shape, especially the error-prone spherical cap and conical frustum.
- Surface area shown alongside volume wherever it's well-defined (sphere, cube, cylinder, cone, box, capsule, cap, frustum, tube, torus).
- Copy the result or download it as a PNG straight from the results panel.
- Input validation with clear error messages — negative dimensions, inner diameter ≥ outer, cap height > diameter, triangle-inequality violations, and self-intersecting torus are all caught.
- Material weight estimator alongside the cost estimator — pick water, concrete, steel, sand, gravel, aluminum, pine wood, or a custom density to convert volume into estimated mass in kilograms and pounds.
- Now covers 15 solids, adding the regular tetrahedron and octahedron (Platonic solids) with their own diagrams, formulas and steps.
- Dedicated partial fill level tool for horizontal cylindrical and spherical tanks — enter the liquid height and instantly see the filled volume and percent full, with full derivation steps.
Frequently asked questions
+What is volume?
Volume is the amount of three-dimensional space a solid occupies. It's measured in cubic units — cubic meters, cubic feet, liters, gallons and so on — because it's a measurement in three dimensions at once.
+What's the formula for the volume of a sphere?
V = (4/3) π r³, where r is the sphere's radius. Doubling the radius makes the volume 8× larger.
+How do I find the volume of a cylinder?
Multiply the area of the circular base by the height: V = π r² h.
+How do I calculate the volume of a cone?
A cone is exactly one-third of the cylinder that would fit around it, so V = (1/3) π r² h with r the base radius and h the perpendicular height.
+What is a spherical cap and how do I find its volume?
A spherical cap is the portion of a sphere cut off by a flat plane. Given the ball radius R and cap height h, V = (π h² / 3)(3R − h). This calculator also lets you enter any two of {ball radius R, base radius a, cap height h} and derives the third from a² = h(2R − h).
+How do I find the volume of a conical frustum?
A frustum is a cone with the top sliced off flat, leaving two circular ends. V = (π h / 3)(r₁² + r₁ r₂ + r₂²), where r₁ and r₂ are the top and bottom radii and h is the height.
+What's the formula for the volume of a torus (donut)?
V = 2 π² R r², where R is the distance from the centre of the torus to the centre of the tube, and r is the tube's radius.
+How many liters are in a cubic meter?
Exactly 1000 liters. And 1 m³ ≈ 264.172 US gallons ≈ 219.969 imperial gallons.
+Can I use this calculator for tank capacity and gallons?
Yes — every result is shown in cubic meters, cubic feet, cubic yards, cubic inches, liters, US gallons and imperial gallons, so you can size fuel tanks, water tanks, aquariums, or fish ponds in whatever unit is convenient.
+How do I estimate concrete or fill volume in cubic yards?
Enter your dimensions in feet or yards, then read the cubic-yards line in the conversion panel. For concrete slabs and footings, the rectangular tank shape (length × width × depth) is usually what you want.
+What units should I enter?
Any consistent length unit — millimeters, centimeters, meters, kilometers, inches, feet or yards. All dimensions on a single calculation must use the same unit, and the answer is in the cube of that unit.
+Can I estimate cost from volume?
Yes — the cost / material estimator that appears with every result multiplies your volume by a price per cubic unit. Useful for concrete, soil, mulch, gravel, resin, water and fill.
+Can I estimate the weight of a material from its volume?
Yes — the material weight estimator next to the cost estimator lets you pick a common material (water, concrete, steel, sand, gravel, aluminum, pine wood) or enter a custom density, then multiplies volume (converted to cubic meters) by density to show mass in kilograms and pounds.
+How do I calculate the volume of a tetrahedron or octahedron?
For a regular tetrahedron with edge length s, V = s³ / (6√2). For a regular octahedron with edge length s, V = (√2 / 3) s³. Both are built into the shape selector above with diagrams and steps.
+How full is a horizontal cylindrical or spherical tank at a given liquid height?
Use the partial fill level tool below. For a horizontal cylinder of radius r and length L filled to height h, V = L r² (θ − sinθ) / 2 where θ = 2 arccos((r−h)/r). For a sphere of radius r filled to height h, V = π h² (3r − h) / 3. Both are shown as a percentage of the tank's total volume.