Spherical sector - area and volume

A spherical sector is the solid cut out of a sphere by a cone whose apex lies exactly at the centre of the sphere. The cone traces a circle on the spherical surface, and everything enclosed between the cone and the piece of sphere it marks off forms the sector. It is therefore built from two familiar bodies: a spherical cap resting on a cone, both sharing the same circular rim of radius a. Two numbers describe it completely — the radius R of the sphere and the height h of the cap measured along the axis. Because every straight segment from the centre to the rim is a radius of the sphere, the slant height of the cone always equals R, which is why the surface formula stays so compact. When h equals R the sector becomes a hemisphere, and when h reaches 2R it fills the entire sphere.

The volume V = (2/3)·π·R²·h depends only on R and h, and the total surface S = π·R·(2h + a) adds the curved cap 2πRh to the lateral surface of the cone πRa. The cap term is also the geometric basis of the solid angle: dividing 2πRh by R² gives the size of the cone of directions in steradians, which is why the shape appears in optics, radiometry and antenna theory. Geographers use the same geometry to measure a polar cap on a globe, engineers meet it in dished vessel ends and dome segments, and in solid geometry the spherical sector is a classic exercise because it combines two simpler solids into one clean pair of formulas.