frustum

A frustum is the portion of a solid that remains between two parallel planes cutting the solid. The two parallel faces are called the bases, and the height is the perpendicular distance between them. When the bases are similar shapes, the frustum is a truncated version of the original solid. The term frustum comes from Latin, meaning a piece or fragment. Frustums occur for cones and pyramids, and are commonly called frustums of a cone or a pyramid.

Frustum of a cone: The solid has two parallel circular bases with radii R and r, and height h. The slant height is s = sqrt(h^2 + (R − r)^2). The volume is V = (h/3) π (R^2 + r^2 + R r). The lateral surface area is L = π (R + r) s, and the total surface area is A = π(R^2 + r^2) + π (R + r) s.

Frustum of a pyramid: The solid has two parallel polygonal bases that are similar, connected by trapezoidal lateral faces. If the base areas are B1 and B2 and the height is h, the volume is V = (h/3) (B1 + B2 + sqrt(B1 B2)). For a right frustum with similar bases, the lateral surface area can be expressed as L = (P1 + P2)/2 × l, where P1 and P2 are the perimeters of the bases and l is the slant height along a lateral face; in general, lateral area is found by summing the areas of the trapezoidal faces.

Frustums are used in architecture, computer graphics, and geometric modeling to describe solids obtained by truncation.