Volume Calculator
Calculate the volume and surface area of 3D geometric shapes
How It Works
Overview
The volume calculator computes the three-dimensional space enclosed by a solid geometric shape, plus its surface area. Pick a shape, enter the relevant dimensions, and you get the volume in cubic units along with the surrounding surface in square units. Use it for tank capacity, packaging, shipping, concrete pours, aquariums, or any 3D measurement task.
Each solid uses a different formula because volumes depend on the shape's cross-section and how it extends through space. Prisms (cube, rectangular box) just multiply base area by height. Pyramids and cones use one-third of the corresponding prism's volume. Spheres and ellipsoids involve π and a (4/3) factor that comes from integration.
The Formula
The formula depends on the solid:
- Cube: V = side³
- Rectangular prism (box): V = length × width × height
- Sphere: V = (4/3) × π × r³
- Cylinder: V = π × r² × h
- Cone: V = (1/3) × π × r² × h (one-third of a cylinder with the same base and height)
- Square pyramid: V = (1/3) × base² × h
- Hemisphere: V = (2/3) × π × r³ (half a sphere)
- Ellipsoid: V = (4/3) × π × a × b × c
Worked Example
Suppose you're sizing a cylindrical hot water tank with a radius of 0.3 m and a height of 1.5 m, and you need its capacity in liters.
- Volume: π × 0.3² × 1.5 = π × 0.135 ≈ 0.4241 m³
- Convert to liters: 0.4241 × 1,000 = 424.1 liters
- Surface area: 2π × 0.3 × (0.3 + 1.5) ≈ 3.39 m²
Compare to a sphere of the same radius: V = (4/3) × π × 0.3³ ≈ 0.113 m³ — only about a quarter of the cylinder's volume because the sphere lacks the vertical extension.
When to Use This
- Tank or container sizing — figure out how many gallons or liters a vessel holds.
- Concrete and material orders — calculate cubic yards needed for a slab, footing, or column.
- Shipping and packaging — confirm a box's volume to compute dimensional weight or fit constraints.
- Aquarium setup — match fish stocking levels to tank capacity (typically 1 inch of fish per gallon).
- Engineering and physics — derive mass from volume × density, or compute buoyancy and displacement.
Common Mistakes to Avoid
- Confusing radius and diameter. Sphere and cylinder formulas need the radius — half the diameter.
- Forgetting the one-third factor. Cones and pyramids hold one-third the volume of the corresponding prism, not the same amount.
- Mismatched units. Mixing meters with centimeters, or feet with inches, gives nonsensical results — convert first.
- Confusing volume with capacity. A thick-walled tank's outer volume can be substantially more than what it can hold internally.
- Reporting in square units. Volume is always cubed (m³, ft³); writing "100 ft²" for a tank's volume is dimensionally wrong.
Frequently Asked Questions
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