What is the relationship between volume and capacity?

Volume is the 3D space a shape occupies (in cubic units like m^3 or ft^3). Capacity is the amount it holds, often in fluid units (liters, gallons). They are the same measurement: 1 liter = 1 cubic decimeter (dm^3) = 1,000 cm^3. 1 US gallon = 231 in^3 = 3.785 liters.

Why is a cone volume one-third of a cylinder?

Archimedes proved this geometrically: any cone fits exactly one-third of the volume of its enclosing cylinder (same base and height). Similarly, a sphere fits exactly two-thirds of its enclosing cylinder. These elegant 1/3 and 2/3 ratios arise from calculus integration.

How do I convert cubic feet to gallons?

1 cubic foot = 7.4805 US gallons. So divide cubic feet by 7.4805 to get US gallons, or multiply by 7.4805 to convert gallons to ft^3. For liters: 1 ft^3 = 28.317 liters.

What is displacement in engines?

Engine displacement is the total volume swept by all pistons (cylinder volume x number of cylinders). A 2.0L engine has cylinders with combined swept volume of 2,000 cm^3 = 2 liters. Larger displacement generally means more power potential.

Volume Calculator — Cubic Volume & Capacity

Volume & Surface Area Formulas

Shape	Volume	Surface Area
Cube	`V = s³`	`SA = 6s²`
Rect. Prism	`V = l×w×h`	`SA = 2(lw+lh+wh)`
Sphere	`V = (4/3)πr³`	`SA = 4πr²`
Cylinder	`V = πr²h`	`SA = 2πr²+2πrh`
Cone	`V = (1/3)πr²h`	`SA = πr²+πrl, l=√(r²+h²)`
Pyramid	`V = (1/3)×base×h`	`SA = base + ½×perimeter×slant`
Torus	`V = 2π²Rr²`	`SA = 4π²Rr`
Capsule	`V = πr²(4r/3+h)`	`SA = 2πr(2r+h)`
Ellipsoid	`V = (4/3)πabc`	`SA ≈ 4π·((aᵖbᵖ+aᵖcᵖ+bᵖcᵖ)/3)^(1/p), p=1.6075`
Frustum	`V = (πh/3)(R²+Rr+r²)`	`SA = π(R²+r²+(R+r)·l), l=√(h²+(R-r)²)`

Shape

Volume

Surface Area

Cube

V = s³

SA = 6s²

Rect. Prism

V = l×w×h

SA = 2(lw+lh+wh)

Sphere

V = (4/3)πr³

SA = 4πr²

Cylinder

V = πr²h

SA = 2πr²+2πrh

Cone

V = (1/3)πr²h

SA = πr²+πrl, l=√(r²+h²)

Pyramid

V = (1/3)×base×h

SA = base + ½×perimeter×slant

Torus

V = 2π²Rr²

SA = 4π²Rr

Capsule

V = πr²(4r/3+h)

SA = 2πr(2r+h)

Ellipsoid

V = (4/3)πabc

SA ≈ 4π·((aᵖbᵖ+aᵖcᵖ+bᵖcᵖ)/3)^(1/p), p=1.6075

Frustum

V = (πh/3)(R²+Rr+r²)

SA = π(R²+r²+(R+r)·l), l=√(h²+(R-r)²)

Mathematically & Metrologically Verified

Last Reviewed: May 2026 | Next Review: May 2027

All formulas, conversion factors, and surface area equations are derived from international standards: ISO 80000-3 (Space and Time), NIST Special Publication 811, and Harris & Stocker, Handbook of Mathematics and Computational Science (Springer, §4.8).

Professional Disclaimer: This tool provides preliminary estimates only. Results do not account for material settling, thermal expansion, or construction tolerance. Consult a licensed engineer before acquiring materials or executing structural work.