Volume Calculator
Calculate volume for 14 common 3D shapes with unit conversion, solve for a missing dimension, build composite volumes (add/subtract a second shape), and optionally compute mass from density — with steps and a shape diagram.
Background
Volume measures how much 3D space something occupies. For lab work, remember: 1 cm³ = 1 mL and 1000 mL = 1 L. Every calculation converts internally to meters and cubic meters, then converts back to whichever units you pick.
How to use this calculator
- Choose a shape and what you want to solve for (volume, or a dimension if supported).
- Pick your length unit and a volume output unit — we'll show equivalents too.
- Optional: enable Composite to add/subtract a second shape, or Mass from density.
- Click Calculate for the answer, steps, and a shape diagram — composite mode also shows a magnitude comparison between the two shapes.
Formulas & Equations Used
Cube: V = a³
Rectangular prism: V = L·W·H
Sphere: V = (4/3)πr³
Hemisphere: V = (2/3)πr³
Spherical cap: V = (1/3)πh²(3R − h)
Ellipsoid: V = (4/3)πabc
Cylinder: V = πr²h
Hollow cylinder (tube): V = πh(R² − r²)
Capsule: V = πr²L + (4/3)πr³
Cone: V = (1/3)πr²h
Conical frustum: V = (1/3)πh(R² + Rr + r²)
Triangular prism: V = (1/2)·b·h·L
Pyramid (rect base): V = (1/3)L·W·H
Truncated pyramid: V = (H/3)(A₁ + A₂ + √(A₁A₂))
Mass from density: m = ρ·V
Composite: Vtotal = V₁ ± V₂
Example Problems & Step-by-Step Solutions
Example 1 — Aquarium volume
- Box dimensions: L=60 cm, W=30 cm, H=40 cm
- Compute volume: V = L·W·H = 60·30·40 = 72,000 cm³
- Convert: 72,000 cm³ = 72,000 mL = 72 L
Example 2 — Beaker height from volume (cylinder)
- Given: V = 250 mL, r = 3.5 cm
- Convert volume: 250 mL = 250 cm³
- Rearrange: h = V/(πr²)
- Compute: h = 250 /(π·3.5²) ≈ 6.50 cm
Example 3 — Tube (hollow cylinder) volume
- Given: outer radius R=2.0 cm, inner radius r=1.5 cm, height h=30 cm
- Use: V = πh(R² − r²)
- Compute: V = π·30·(2² − 1.5²) = π·30·(4 − 2.25) = π·52.5 ≈ 164.9 cm³
- Convert: 164.9 cm³ ≈ 164.9 mL
Example 4 — Composite volume: block with a drilled hole
- Block: L=10 cm, W=8 cm, H=6 cm → V₁ = 480 cm³
- Drilled cylindrical hole: r=1 cm, h=6 cm → V₂ = π·1²·6 ≈ 18.85 cm³
- Subtract: Vtotal = V₁ − V₂ = 480 − 18.85 ≈ 461.15 cm³
Example 5 — Mass of water from volume and density
- Cylinder: r=5 cm, known volume V=500 mL
- Solve for height: h = V/(πr²) = 500/(π·5²) ≈ 6.37 cm
- Density of water: ρ = 1 g/mL = 1000 kg/m³
- Mass: m = ρ·V = 1000 × 0.0005 = 0.5 kg = 500 g
Frequently Asked Questions
Q: Is cm³ the same as mL?
Yes. 1 cm³ = 1 mL.
Q: Can I solve for height or radius instead of volume?
Yes — use the Solve for dropdown to compute a missing dimension. Available options depend on the shape; some shapes (like a spherical cap) only support solving for volume, since solving for their height would require a more complex equation than a simple rearrangement.
Q: What does "composite volume" mean?
It means adding or subtracting the volumes of two shapes — for example, a rectangular block with a cylindrical hole drilled through it.
Q: Why do some shapes only let me solve for one dimension, not all of them?
A few volume formulas (like the spherical cap or truncated pyramid) can't be cleanly rearranged for every dimension using basic algebra — some would need solving a cubic equation or similar. This calculator only offers a "solve for" option where the rearranged formula is exact and reliable.
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