Spherical Cap Calculator

What does this spherical cap calculator compute?

A spherical cap of height h on a sphere of radius r, with 0<h≤2r, has volume (1/3)πh²(3r−h), base radius √(h(2r−h)), curved area 2πrh, and total surface 2πrh+πa². h=r is a hemisphere. h=2r is the full sphere.

The published formula or method is V = (1/3)πh²(3r−h), A = 2πrh + πa². The following references support the stated mathematics or method:

• Gilbert Strang, Calculus — Chapter 8: Applications of the Integral (MIT OpenCourseWare): 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.

Which inputs and limits should you check?

Lengths, areas, and volumes have different units. Check whether a reported area is curved/lateral or total and use the shape-specific dimension constraints.

The worked example uses r = 1, h = 1 with V = 2π/3. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.geometry.solid v1.19.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical scope and complete results

Use one consistent length unit: lengths are in L, surface areas in L² and volumes in L³. Inputs are finite decimal numbers or strings. Positive dimensions and nonzero output measures must fit normal IEEE-754 binary64. NUMERIC_UNDERFLOW rejects raw nonzero text that converts to zero; NUMERIC_RANGE asks you to rescale when a required measure is outside that range. Outputs retain binary64 precision, without a small-value zero clamp. Input rounding affects very thin shapes; this is numerical evaluation, not exact decimal arithmetic. Regular n must be a safe integer from 3 to 9007199254740991. Ellipsoid minimum/maximum semi-axis ratio must be at least 1e-100.

The 2-D drawing shows a section, elevation or face, not a 3-D mesh. Very thin shapes, tiny or almost full angles and polygons above 128 sides omit the drawing while retaining complete numerical results. Platonic height denotes vertex height for a tetrahedron, opposite-vertex distance for an octahedron, and circumsphere diameter for an icosahedron or dodecahedron. A spherical lune is a surface; its auxiliary volume is the associated wedge volume.

Ordinary interactive calculation runs locally. Shared URLs send their parameters to the server; REST and MCP send inputs for execution. Sharing does not recreate a historical engine: a version mismatch shows the current-engine notice. CSV and result JSON include all calculated fields and raw provided inputs.