Spherical sector calculator

A spherical sector, sometimes called a spherical cone, is the solid formed when a sphere is cut by a cone whose apex sits at the sphere center and whose base is the rim of a spherical cap. Two positive lengths fix the figure completely: the sphere radius R and the cap height h measured along the axis from the cap base plane to the sphere surface. Once those are known and the height does not exceed the sphere diameter, the classical closed forms give the sector volume as two-thirds pi times R squared times h, the circular rim radius as the square root of h times two-R-minus-h, and the total surface area as pi times R times the sum of twice the height and that rim radius. Students meet these identities in solid-geometry units that follow sphere volume and spherical segments; engineers and designers need the same numbers when a dome, tank head, or packaging sketch lists radius and dome height rather than a full CAD solid.

What a spherical sector is and which two measures determine it

In Euclidean solid geometry a spherical sector is the region swept by radii from the sphere center out to every point of a spherical cap. Equivalently, it is the union of that cap with the right circular cone whose apex is the center and whose base is the circular rim of the cap. The sphere radius R is the distance from center to surface; the cap height h is the sagitta from the plane of the rim to the cap vertex along the symmetry axis.

The rim radius follows at once from the right triangle in an axial cross-section: a equals the square root of h times two-R minus h. The sector volume formula V equals two-thirds pi R squared h is standard in textbooks and reduces correctly when h equals two R: the sector fills the whole ball and the volume becomes four-thirds pi R cubed.

The total surface area counts both the spherical zone of the cap, which contributes two pi R h, and the lateral surface of the cone, which contributes pi a R, combining into pi R times two h plus a. Search intent clusters around phrases like spherical sector calculator, volume of a spherical sector, spherical cone volume, sector surface area from radius and height, and cap height twice radius limit.

People want numeric results they can check against an answer key or a tank sketch, not only a geometric argument that the solid is determined.

How this calculator validates input and applies the closed-form formulas

Each must be a positive finite number. When the bound holds, base radius a is the square root of height times two-radius minus height, volume is two-thirds pi times radius squared times height, and surface area is pi times radius times the sum of twice the height and a. Edge cases are explicit: missing fields, non-numeric strings, zero or negative radius, zero or negative height, and height larger than twice the radius all fail before any metric is returned.

The boundary h equals two R is accepted and yields the full-sphere volume with a degenerate zero rim radius, matching the classical limit. You may use any consistent unit system for radius and height; volume and area are expressed in the corresponding cubic and square units without conversion inside the engine.