Halfspheres

Half-spheres, or hemispheres, are geometric shapes formed by dividing a sphere into two equal parts with a plane that passes through the sphere’s center. The term is often used to refer to either the solid half-ball or the curved half-spherical surface, with the distinction that the plane cut may or may not be considered part of the boundary.

Two common interpretations exist. A solid hemisphere is the half of the solid ball, including the base disk formed by the cut. A hemispherical surface is the curved outer surface of that half, excluding the base. When the plane cut is included, the base disk contributes to the boundary of the solid hemisphere.

For a sphere of radius r, the solid hemisphere has volume 2/3 π r^3. The curved surface area is 2π r^2, and the total surface area of the solid hemisphere (including the base disk) is 3π r^2. The height from the base to the pole is r, and the base is a circle of radius r.

In higher dimensions, the concept generalizes to an n-hemisphere, formed by halving an n-sphere with a hyperplane through the center. In n dimensions, many measures of the hemisphere are half of the corresponding measures for the full n-ball, with analogous boundary considerations.

Applications of hemispheres appear in architecture, physics, computer graphics, and mathematical analysis, where they model half-space regions, assist with integrals over hemispherical domains, or serve as convenient geometric references for symmetry and contact problems.