Prismatoid
A prismatoid is a polyhedron all of whose vertices lie in two parallel planes, with the faces contained in these planes serving as the bases and the remaining lateral faces connecting the bases.[1] The perpendicular distance between the two parallel planes is known as the altitude or height of the prismatoid.[1] Lateral faces are typically triangles, trapezoids, or parallelograms, formed by edges linking vertices between the bases.[2] Prismatoids encompass a variety of common polyhedral solids, including prisms (where the bases are congruent and parallel polygons), pyramids (where one base degenerates to a single point), and frustums of pyramids or cones (truncated versions with polygonal bases).[2] They can be convex, as in the case of the convex hull of two polygons in parallel planes, and extend to smooth variants defined by curves rather than polygons in computational geometry contexts.[3] These solids are fundamental in solid mensuration and appear in applications such as earthwork volume calculations in civil engineering.[4] The volume of a prismatoid is computed using the prismoidal formula, which states that V = \frac{h}{6} (B_1 + B_2 + 4M), where h is the height, B_1 and B_2 are the areas of the two bases, and M is the area of the midsection (the cross-section parallel to the bases at the midpoint).[1] This formula provides an exact volume for prismatoids and approximates volumes for more general solids where cross-sectional areas vary quadratically with height.[2] For specific cases like prisms, the formula simplifies to V = h \cdot B, where B is the base area, while for pyramids it reduces to V = \frac{1}{3} h B.[2]Definition and Terminology
Formal Definition
A prismatoid is defined as a polyhedron in which all vertices lie in exactly two distinct parallel planes.[5][6] The two faces formed by the intersections of these planes with the polyhedron serve as the bases, which are simple polygons that need not be congruent or regular.[5][7] The lateral faces of a prismatoid connect corresponding edges of the two bases and are either trapezoids, when the bases have the same number of sides, or triangles, in cases where one base degenerates to a point (as in pyramids).[5][2] This structure visually resembles a prism with parallel and equal bases but generalizes to allow unequal bases, providing flexibility in connecting polygonal outlines across the parallel planes.Etymology and History
The term "prismatoid" derives from the Greek prīsmat- (stem of prîsma, meaning "prism" or "something sawed") combined with the suffix "-oid," indicating resemblance, thus describing a polyhedral form akin to a prism.[8][9] The earliest recorded use of the term in English dates to 1858, attributed to Robert Mayne in his expository lexicon.[8] By the late 19th century, "prismatoid" had entered mathematical discourse more widely, appearing in George Bruce Halsted's Metrical Geometry: An Elementary Treatise on Mensuration in 1881 and in encyclopedic references circa 1890.[10][11] This development occurred amid 19th-century advancements in polyhedral classification, where the prismatoid concept addressed solids confined between parallel planes, extending foundational ideas from prisms studied in earlier geometry.[12]Geometric Structure
Bases and Lateral Faces
A prismatoid is defined as a polyhedron with two polygonal bases lying in parallel planes, which may have different numbers of sides, such as a triangular base connected to a hexagonal one.[5] These bases form the foundational parallel faces of the structure, with all vertices confined to these two planes. The bases are polygons, typically convex.[5] The lateral faces of a prismatoid connect the edges or vertices between the two bases, typically consisting of trapezoids or triangles.[13] If the bases have the same number of sides and corresponding edges align, the lateral faces are quadrilateral trapezoids, as seen in frustums where each trapezoid spans parallel edges from both bases.[14] In cases where one base has fewer vertices, such as a pyramid-like configuration with an apex base, the lateral faces become triangles, each connecting an edge of the larger base to a vertex of the smaller one.[5] Connectivity between bases and lateral faces follows strict rules to maintain the polyhedron's integrity, with each lateral face spanning directly from an edge or vertex of one base to the other without crossing edges.[7] This arrangement typically results in a convex prismatoid when the bases are convex polygons, with the lateral faces adjoining sequentially around the perimeter of the bases, forming a continuous band that links the two parallel polygons; non-convex variants also exist.[14][7]Vertex and Edge Configurations
In a prismatoid, all vertices are partitioned between two distinct parallel planes, with no vertices located elsewhere in space, ensuring the polyhedron's structure is confined to these bounding planes.[5][2] One plane contains the vertices of the first base polygon, while the other holds those of the second base polygon, allowing for flexible arrangements such as a triangular base with three vertices opposite a rectangular base with four vertices.[15] The edges of a prismatoid fall into two primary categories: base edges, which lie entirely within one of the parallel planes and form the boundaries of the polygonal bases, and lateral edges, which connect vertices between the two planes and may be slanted rather than perpendicular.[5][15] Base edges outline the polygons in each plane, while lateral edges bound the triangular or trapezoidal faces that link the bases, with each such edge spanning from a vertex in one plane to a vertex in the other.[2] Configurations of vertices and edges in prismatoids permit arbitrary numbers of vertices per base, typically at least three for each polygonal base when assuming convex polygons.[15] The total number of edges equals the sum of the base edges from both polygons plus the lateral edges, where the latter depend on the specific connections required to form the enclosing faces without gaps or overlaps; for instance, a configuration with m vertices on one base and n on the other yields m + n base edges plus at least m + n lateral edges in minimal convex hulls.[5][15] From a graph-theoretic perspective, the 1-skeleton of a prismatoid consists of two disjoint polygonal graphs for the bases connected by a simple bipartite graph of lateral edges between the vertex sets of the two planes.[15] For validity as a polyhedron, these vertices and edges must assemble into a simply connected structure without self-intersections, ensuring the structure remains a well-defined, non-degenerate solid.[2]Volume Calculation
Prismatoid Volume Formula
The volume of a prismatoid is given by the formulaV = \frac{h}{6} (A_1 + 4A_m + A_3),
where h is the perpendicular distance between the two parallel planes containing the bases, A_1 and A_3 are the areas of the respective bases, and A_m is the area of the mid-parallel cross-section.[5][2][7] The term A_m refers to the area of the polygonal section parallel to the bases and located at a height of h/2 from each base. This midsection arises from the linear interpolation of the positions between the bases, forming a polygon whose vertices are determined by averaging the coordinates of corresponding vertices from the two bases or by projecting vectors along the lateral edges to the midway plane.[5][1][2] In special cases, the formula simplifies accordingly. If the bases are congruent and equal in area (A_1 = A_3 = A_m = A), the prismatoid reduces to a prism with volume V = h A. If one base has zero area (e.g., A_1 = 0), it becomes a pyramid with base area A_3 and volume V = \frac{1}{3} h A_3, assuming the midsection area A_m = A_3 / 4 due to the quadratic scaling of cross-sectional areas in a pyramid.[5][16][1] To compute A_m practically, identify corresponding vertices on the bases—typically aligned via the trapezoidal or triangular lateral faces—and calculate their midpoint positions in the midway plane, then determine the area of the resulting polygon using standard polygonal area methods, such as the shoelace formula or decomposition into triangles. Alternatively, vector projections from base vertices to the mid-plane can yield the same cross-section for verification.[2][1]