frusta

Frusta are the portions of geometric solids that remain after a plane cuts the solid with two parallel planes, removing a top part. The term is commonly used for the parts of cones or pyramids. The plural form frusta is standard in mathematics, though frustums is also common.

In a cone frustum, the two parallel bases are circles with radii R and r, and the height between the bases is h. The volume is V = (1/3) π h (R^2 + R r + r^2). The lateral surface area is L = π (R + r) s, where s is the slant height given by s = sqrt(h^2 + (R − r)^2). The total surface area adds the areas of the two bases: A = π (R^2 + r^2) + π (R + r) s.

In a pyramidal frustum, formed by slicing a pyramid with a plane parallel to its base, the two bases are similar polygons with areas B1 and B2 and the height between bases is h. The volume is V = h/3 (B1 + B2 + sqrt(B1 B2)). The lateral surface consists of trapezoidal faces and depends on the shape of the base polygon; it is typically computed by summing the areas of those faces.

Etymology and usage: the word frustum derives from Latin and describes a consumer of a larger solid into a smaller, truncated form. In English, frusta is widely used as the plural for frustum, especially in mathematical contexts.

Frusta appear in various applications, including architectural fragments, funnels, and components in engineering models, where truncation of shapes is a practical approximation.