- ISO 80000-2:2019 Quantities and units — Mathematics
Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
- NIST DLMF §19.2 Incomplete elliptic integrals — 19.2(ii) Legendre’s integrals F(φ,k) and E(φ,k)
Supports: Ellipsoid triaxial surface uses incomplete F and E with modulus k, not parameter m = k²
- Euclid’s Elements — XIII.13: Tetrahedron
Supports: Construction and edge-to-circumsphere geometry of the regular tetrahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
- Euclid’s Elements — XIII.14: Octahedron
Supports: Construction and edge-to-circumsphere geometry of the regular octahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
- Euclid’s Elements — XIII.16: Icosahedron
Supports: Construction and edge-to-circumsphere geometry of the regular icosahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
- Euclid’s Elements — XIII.17: Dodecahedron
Supports: Construction and edge-to-circumsphere geometry of the regular dodecahedron. Face areas are obtained from triangle or regular-pentagon geometry; volume is the sum of pyramids from the center to the faces. These area and volume expressions are derived from the construction and the separately cited pyramid-volume method. Mathematical reference only; not validation or endorsement of this implementation.
- NIST DLMF — 19.33 Triaxial Ellipsoids — 19.33(i), equations 19.33.1–19.33.3
Supports: Ellipsoid volume V = 4πabc/3 and exact surface-area expressions in symmetric and Legendre elliptic integrals. An approximate displayed area, where offered, must be distinguished from these exact identities. Mathematical reference only; not validation or endorsement of this implementation.
- Gilbert Strang, Calculus — Chapter 8: Applications of the Integral (MIT OpenCourseWare) — §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §8.5 Masses and Moments, exercises 24–29 §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area §§8.1–8.3 Areas and Volumes by Slices, Curve Length and Surface Area
Supports: Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Pappus’s volume and surface-area theorems, with a torus volume exercise. For a disk of radius r whose center is distance R>r from the axis, V = 2π²Rr²; for its boundary circle the surface theorem gives S = 4π²Rr². These expressions apply the stated centroid theorems. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX. Cross-section integration V = ∫A(z) dz and the surface-of-revolution integral ∫2πf(x)sqrt(1+f′(x)²) dx, with sphere, cone and cylinder applications. Constant sections give prism volume; similar sections give pyramid/frustum volume; integrating a circular profile gives cap area and volume, and differences of caps give spherical zones. Flat end faces are added separately where specified. These composite formulas are derived applications, not individually tabulated implementation checks. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
- Gilbert Strang, Calculus — Chapter 1: Introduction to Calculus (MIT OpenCourseWare) — §1.5 A Review of Trigonometry
Supports: Right-triangle definitions of sine, cosine and tangent, the sine/cosine identities, and the law-of-cosines and parallelogram exercises 31–32. Chord length follows by bisecting the radii triangle. Regular-polygon apothems and areas follow by splitting the polygon into equal triangles. The law of sines follows by equating a triangle’s altitude computed from either adjacent side; SSA can give two triangles. These applications are derived from the stated relations. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
- Gilbert Strang, Calculus — Chapter 14: Multiple Integrals (MIT OpenCourseWare) — §14.4 Cylindrical and Spherical Coordinates
Supports: Spherical coordinates and volume element ρ² sin(φ) dρ dφ dθ. Restricting azimuth to α gives a wedge α/(2π) of a ball and the corresponding surface lune. Radial integration to r with polar angle φ₀ and h=r(1−cos φ₀) gives spherical-sector volume 2πr²h/3. These formulas are derived applications; cap and cone surfaces use the separate surface reference. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.
