Volume Calculator

Using this calculator is straightforward. First, click the shape button that matches your 3D solid. The dimension inputs update automatically to show only the fields relevant to that shape — so for a sphere you only see Radius (r), while for a rectangular prism you see three separate dimension fields.

Enter each dimension as a positive number (decimals are supported). Click Calculate and the results panel instantly shows the volume in cubic units and the surface area in square units, together with the formulas used and a complete step-by-step working breakdown.

Use the Presets to load a worked example for any shape with a single click. These preset values use clean numbers that demonstrate the formulas clearly and make it easy to check your own calculations.

All units are dimensionless — the calculator works with any unit system (centimetres, metres, inches, feet). If your inputs are in centimetres, the volume is in cubic centimetres (cm³) and the surface area is in square centimetres (cm²).

Surface area is the total area of all outer faces of a 3D shape, measured in square units. It matters for real-world applications such as calculating the amount of material needed to wrap, coat, or manufacture an object.

For prisms and cylinders, surface area = lateral face area + 2 × base area. For cones and pyramids, surface area = lateral face area + base area, where the lateral area depends on the slant height l (the distance from the apex to the base edge — not the perpendicular height h).

The torus surface area formula SA = 4π²Rr is derived directly from Pappus's centroid theorem: revolve a circle of circumference 2πr through a distance of 2πR, giving (2πr)(2πR) = 4π²Rr. For the ellipsoid, no closed-form exact formula exists, so this calculator uses Thomsen's approximation with exponent p = 1.6075, which has a maximum error below 1.061% — accurate enough for any practical engineering or academic use.

The capsule combines a cylinder with two hemisphere caps. Since two hemispheres make one complete sphere, the surface area simplifies to: SA = 2πr(2r + h) = cylinder lateral area + sphere area.

Always use consistent units throughout a calculation. If your radius is in metres, your height must also be in metres — the resulting volume will be in cubic metres. Converting between units after calculating is safer than mixing units during calculation.

For the cone, the height h is the perpendicular height from the base to the apex, not the slant height. The calculator derives slant height internally as l = √(r² + h²).

For the torus, the minor radius r₂ must be strictly less than the major radius R. If r₂ ≥ R, the tube self-intersects and the torus formula no longer applies. The calculator validates this condition.

For the triangular prism, this calculator assumes a right triangle cross-section with legs a (base) and b (height). The hypotenuse is computed automatically. For a general (non-right) triangular prism, use the area & perimeter tool to compute the triangle area separately.