What is volume?
Volume is the amount of 3D space a solid occupies, measured in cubic units (cm³, m³, ft³, in³). For most familiar shapes, volume reduces to a single formula based on the shape's defining dimensions: a side length, a radius, a height, or some combination.
The Volume Calculator handles the six most common shapes in one tool — pick the shape, enter the dimensions, get the answer. Each shape has its own formula:
- Cube: V = s³
- Rectangular box: V = l × w × h
- Cylinder: V = π × r² × h
- Sphere: V = (4/3) × π × r³
- Cone: V = (1/3) × π × r² × h
- Square pyramid: V = (1/3) × s² × h
How to use the volume calculator
- Pick the shape from the dropdown.
- The input fields adapt — one field for cube and sphere, two for cylinder/cone/pyramid, three for rectangular box.
- Enter your dimensions in any unit, but be consistent across all fields for a given shape.
- The volume appears instantly with the formula displayed.
- For shape-specific extras (cylinder lateral area, sphere surface, cone slant height), use the dedicated single-shape calculators linked in the sidebar.
Worked examples
Example 1 — Cube, s = 5
V = 5³ = 125 cubic units. A cube with 5 cm sides holds 125 cm³, just over an eighth of a litre.
Example 2 — Rectangular box, 10 × 6 × 4
V = 10 × 6 × 4 = 240. A 10 × 6 × 4 inch box holds 240 in³ ≈ 4 litres.
Example 3 — Cylinder, r = 3, h = 10
V = π × 9 × 10 ≈ 282.74. A cylinder 3 cm radius × 10 cm tall holds about 283 mL — roughly the volume of a soda can (which is sized close to this).
Example 4 — Sphere, r = 5
V = (4/3) × π × 125 ≈ 523.6. A sphere with 5 cm radius holds about half a litre — close to a tennis ball volume.
Example 5 — Cone, r = 4, h = 12
V = (1/3) × π × 16 × 12 ≈ 201.06. An ice-cream cone shape with 4 cm radius and 12 cm tall holds about 200 mL — a generous scoop's worth of melted ice cream.
Example 6 — Square pyramid, s = 6, h = 9
V = (1/3) × 36 × 9 = 108. A square pyramid with 6 cm base sides and 9 cm tall holds 108 cm³.
The factor-of-1/3 pattern
You'll notice that cones and pyramids both have a "(1/3)" in their formulas. That's not a coincidence — it's a general fact about "pointy" shapes vs "straight" shapes:
- A cone is 1/3 of the cylinder that fits around it (same radius, same height).
- A square pyramid is 1/3 of the rectangular box that contains it (same base, same height).
- Any pyramid (with any base shape) is 1/3 × base area × height.
The 1/3 is provable with calculus or with a physical demo — fill a hollow cone three times to fill the matching cylinder. The relationship was known to Greek mathematicians: Eudoxus first proved it; Archimedes generalized it.
The (4/3)π for spheres
The sphere's coefficient (4/3)π takes more work to derive. It comes from integrating cross-sectional discs from -r to +r. Geometrically, Archimedes showed that the sphere takes exactly 2/3 of the smallest cylinder that fits around it. Cylinder volume: πr² × 2r = 2πr³. Sphere = 2/3 × 2πr³ = (4/3)πr³.
This was Archimedes' favorite result — he asked for a sphere-inside-cylinder diagram to be carved on his tombstone. Two thousand years later, Cicero rediscovered the tomb in Sicily by looking for that exact carving.
Units and conversions
Volume always uses cubic units. The cube of the input unit:
- Input cm → cm³ (1 cm³ = 1 mL)
- Input m → m³ (1 m³ = 1000 L)
- Input ft → ft³ (1 ft³ ≈ 7.48 US gallons ≈ 28.32 L)
- Input in → in³ (1 in³ ≈ 16.39 cm³)
For cooking and chemistry: 1 cm³ = 1 mL exactly (by definition of the litre). 1000 cm³ = 1 L. 1 m³ = 1000 L = 1 cubic metre. Converting between metric and imperial requires the standard conversions (a cubic foot is about 28 litres, a cubic metre is about 35 cubic feet).
Why the cube-square law matters
For all of these shapes, volume scales with the cube of the linear dimension while surface area scales with the square. Doubling all dimensions makes the volume 8× bigger but the surface area only 4× bigger. This has profound real-world consequences:
- Heat regulation in animals: bigger animals have proportionally less surface area per unit of mass, so they retain heat better. Big mammals can live in cold climates; tiny mammals can't.
- Drug dosing in children: dose scales roughly with surface area (which is closer to mass^(2/3)) rather than mass directly, because metabolic rate is surface-area-limited.
- Industrial chemistry: scaling up a 1L lab reaction to a 1000L production reactor is non-trivial because heat transfer (a surface phenomenon) doesn't keep pace with reaction rate (a volume phenomenon).
- Storage efficiency: bigger storage tanks are more material-efficient per litre stored — this is why fuel tanks and water towers are typically as large as logistics permit.
Common shapes you might combine
The calculator handles single shapes; for combined shapes, decompose and add:
- Capsule (pill shape) = cylinder + 2 hemispheres = cylinder + 1 full sphere of the same radius.
- Cylinder with conical end = cylinder volume + cone volume (using cone height where the cylinder ends).
- Hollow cylinder (pipe) = outer cylinder volume − inner cylinder volume.
- Truncated cone (frustum) = full cone volume − small cone volume on top.
- Hollow sphere (shell) = outer sphere − inner sphere.
Where you actually use volume calculations
Containers and packaging
"Will this fit in that box?" — compare the volumes. A 10 × 10 × 10 in box holds 1000 in³ ≈ 16.4 L. A laptop with a 1.5 L volume fits comfortably; a 20 L lamp does not.
Cooking and brewing
Recipe scaling: a recipe for a 9-inch round pan needs to scale by the volume ratio when you change to a 12-inch round pan. 9-inch round (1.5-inch deep): π × 4.5² × 1.5 ≈ 95 in³. 12-inch round (1.5-inch deep): π × 6² × 1.5 ≈ 170 in³. So you'd multiply the recipe by 170/95 ≈ 1.79.
Aquariums and pools
Aquarium volume = length × width × depth (rectangular box). A 30 × 12 × 18 inch tank holds 6,480 in³ = about 28 US gallons. Pool volume similarly: a 30 ft × 15 ft × 5 ft (avg depth) pool = 2250 ft³ ≈ 16,800 US gallons.
Construction estimates
Concrete is sold by the cubic yard (or cubic metre outside the US). A 10 ft × 10 ft slab at 4 inches thick: 10 × 10 × (4/12) = 33.3 ft³ = 1.23 cubic yards. Order at least 1.5 yards to allow for spillage and depth variation.
Material orders
Mulch, gravel, soil, sand — all sold by volume. Calculate the volume of the area you're filling (length × width × depth), order at least 10% extra.
Industrial and scientific work
Tank capacities, reaction-vessel sizes, fuel-system volumes, gas storage. The same six formulas drive everything from laboratory glassware specifications to refinery storage tank engineering.
Common mistakes
- Forgetting the (1/3) for cones and pyramids. πr²h is the cylinder formula; cones are 1/3 of that. Same for square base × height vs pyramids.
- Forgetting the (4/3) for spheres. πr³ is wrong by a factor of 4/3. The (4/3) is real and necessary.
- Mixing units. Length in feet × width in inches gives nonsense. Convert all dimensions to one unit first.
- Using diameter where radius should go. The cylinder, cone, and sphere formulas all use radius (half the diameter). Halve the diameter before plugging in.
- Confusing surface area with volume. Surface area (in squared units) is for paint, wrap, or insulation. Volume (in cubed units) is for capacity, fill, or weight. Different formulas, different units.
What the calculator gives you, summarized
- Six shapes in one tool — cube, rectangular box, cylinder, sphere, cone, square pyramid.
- Adaptive inputs — fields change to match the selected shape's required dimensions.
- Formula display — the formula and your numbers plugged in are shown alongside the result.
- Cubic-units reminder — output is in whatever unit you input, cubed.
- Specialized tools linked — for shape-specific extras (slant height, surface area details), the dedicated single-shape calculators handle those.
One picker, six shapes, instant volume. The geometry-class staple, in one tool.