Mathematical Methodology & Standards
Complete transparency into the classical geometry equations, calculus integrations, physical conversion ratios, and material constants used across our 14 calculators.
Classical 3D Geometric Formulas
Right Circular Cylinder
Where r is internal radius and h is total height or length.
Sphere
Full spherical volume; also supports spherical caps via V_cap = (πh²/3)(3r - h).
Cone & Truncated Frustum
For a standard cone with top radius r = 0, reduces to V = (1/3)πR²h.
Pyramid (Rectangular Base)
Base area multiplied by perpendicular height, divided by three.
Triangular Prism
Cross-sectional triangle area (0.5 × b × h) projected along prism length L.
Concrete Slab & Rectangular Prism
Automatic conversion from feet/inches to cubic yards (yd³ = ft³ / 27).
Horizontal Tank Partial-Fill Integration
Calculating the partial liquid volume in a horizontal cylinder at liquid depth h requires evaluating the cross-sectional area of a circular segment:
Where r is radius, h is fluid fill depth, and the angle is evaluated in radians. For tanks with hemispherical or 2:1 semi-elliptical heads, volume is evaluated as a spherical/ellipsoidal cap addition.
NIST Handbook 44 Conversion Standards
All multi-unit output matrices utilize verified physical conversion factors defined by the National Institute of Standards and Technology (NIST SP 811) and the International System of Units (SI).
| Unit Name | Exact Statutory Definition |
|---|---|
| 1 US Liquid Gallon | 231.0 cubic inches = 3.785411784 liters |
| 1 Imperial Gallon (UK) | 4.54609 liters = 1.20095 US gallons |
| 1 Cubic Yard (yd³) | 27 cubic feet = 764.554857984 liters = 0.76455 m³ |
| 1 Cubic Foot (ft³) | 1,728 cubic inches = 7.48052 US liquid gallons |
| 1 Cubic Meter (m³) | 1,000 liters = 35.3147 cubic feet = 1.30795 yd³ |
| 1 Oil Barrel (bbl) | 42 US gallons = 158.9873 liters |
Reference Material Densities for Mass Computations
When calculators estimate total weight from volume (e.g. Water Volume Calculator, Concrete Volume Calculator), these empirical densities are applied:
| Material | Metric Density | Imperial Density | Engineering Notes |
|---|---|---|---|
| Water (Fresh @ 4°C) | 1,000 kg/m³ | 62.43 lb/ft³ | Standard NIST fluid density |
| Normal-Weight Concrete | 2,400 kg/m³ | 149.8 lb/ft³ | 4,000 lb/yd³ standard ready-mix |
| Crushed Stone / Gravel | 1,600 kg/m³ | 99.9 lb/ft³ | Compacted base aggregate |
| Dry Sand | 1,602 kg/m³ | 100.0 lb/ft³ | Settled builder sand |
| Diesel Fuel (#2) | 832 kg/m³ | 51.9 lb/ft³ | 6.94 lb/gal @ 15°C |
Medical & Specialized Formulas
ARDSNet Predicted Body Weight (PBW)
Target tidal volume is titrated to 4–8 mL/kg of predicted body weight to mitigate ventilator-induced lung injury (VILI).
Prostate Ellipsoid Formula (TRUS/MRI)
Standard prolate ellipsoid formula applied to ultrasound transverse and sagittal gland dimensions.