Volume Calculator

Euclidean Solid Geometry & 3D Mensuration Standards

Solid geometry provides closed-form equations for calculating the enclosed 3D space (volume) and boundary enclosing surface area (A_surface).

Primary 3D Shape Formulas

• Cylinder: V = π r^2 h, A_total = 2π r

• Sphere: V = (4) / (3) π r^3, A = 4 π r^2

• Cone: V = (1) / (3) π r^2 h, A = π r

• Rectangular Box / Prism: V = l · w · h, A = 2

• Cube: V = a^3, A = 6a^2

• Square Pyramid: V = (1) / (3) a^2 h, A = a^2 + 2a√(^2 + h^2)

• Torus: V = 2π^2 R r^2, A = 4π^2 R r

Liquid Capacity Conversions

• 1 US Gallon = 231 in^3 = 0.13368 ft^3 = 3.78541 Liters

• 1 Liter = 1,000 cm^3 = 0.001 m^3 = 61.0237 in^3

How to Use This Calculator

Enter your 3D Geometric Solid Shape — choose from Cylinder (Radius r, Height h), Sphere (Radius r), Rectangular Box / Prism (Length l, Width w, Height h), Dimension A (Radius r / Length l / Side a / Major R) (e.g. 5.0), Dimension B (Height h / Width w / Minor r / Semi-axis b) (e.g. 10.0), Dimension C (Height h for Box / Semi-axis c for Ellipsoid) (e.g. 8.0), and Measurement Unit — choose from Inches (in), Feet (ft), Centimeters (cm). Results update instantly as you type — adjust any value to compare scenarios.

The Formula

Formula Reference: Volume Calculator

Input variables:

solidShapeSelect — 3D Geometric Solid Shape

dimensionA — Dimension A (Radius r / Length l / Side a / Major R)

dimensionB — Dimension B (Height h / Width w / Minor r / Semi-axis b)

dimensionC — Dimension C (Height h for Box / Semi-axis c for Ellipsoid)

unitOfMeasure — Measurement Unit

Computed outputs:

calculatedVolume — Calculated 3D Volume

totalSurfaceArea — Total Surface Area

equivalentLiters — Liquid Volume (Liters)

equivalentGallons — Liquid Volume (US Gallons)

lateralSurfaceArea — Lateral / Curved Surface Area

Mathematical relationships extracted from calculation logic:

vol = Math.PI Math.pow(a, 2) b

lateralArea = 2 Math.PI a * b

surfArea = lateralArea + (2 Math.PI Math.pow(a, 2))

slant = Math.sqrt(Math.pow(a, 2) + Math.pow(b, 2))

slantH = Math.sqrt(Math.pow(a / 2, 2) + Math.pow(b, 2))

liters = vol * 0.016387064

gallons = vol * 7.48051948

Reference constants:

p = 1.6075

Standard: NIST / ISO mathematical definitions

Worked Example

A cylindrical tank measuring 5 inches in radius and 10 inches high holds 785.4 cubic inches of liquid, which equals exactly 3.40 US gallons or 12.87 liters.

Understanding Your Result

Geometric calculations assume ideal shapes — real-world measurements should include a 5–15% tolerance for irregular shapes, cutting waste, and installation overlap. For construction applications: always verify calculated areas match your physical measurements before ordering materials. Standard practice is to measure the same dimension twice from different reference points and use the average. Express results in the units required by your supplier or building permit — conversions between cubic yards, cubic feet, and cubic meters must be precise for material orders.