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In geometry, a triangular prism or trigonal prism [1] is a prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a right triangular prism. A right triangular prism may be both semiregular and uniform. The triangular prism can be used in constructing another polyhedron.
To calculate the formula for the surface area and volume of a gyrobifastigium with regular faces and with edge length , one may adapt the corresponding formulae for the triangular prism. Its surface area A {\displaystyle A} can be obtained by summing the area of four equilateral triangles and four squares, whereas its volume V {\displaystyle V ...
An elongated triangular orthobicupola with a given edge length has a surface area, by adding the area of all regular faces: [2] (+). Its volume can be calculated by cutting it off into two triangular cupolae and a hexagonal prism with regular faces, and then adding their volumes up: [2] (+).
The Egyptians knew the correct formula for the volume of such a truncated square pyramid, but no proof of this equation is given in the Moscow papyrus. The volume of a conical or pyramidal frustum is the volume of the solid before slicing its "apex" off, minus the volume of this "apex":
An oblique prism is a prism in which the joining edges and faces are not perpendicular to the base faces. Example: a parallelepiped is an oblique prism whose base is a parallelogram, or equivalently a polyhedron with six parallelogram faces. Right Prism. A right prism is a prism in which the joining edges and faces are perpendicular to the base ...
If the areas of the two parallel faces are A 1 and A 3, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A 2, and the height (the distance between the two parallel faces) is h, then the volume of the prismatoid is given by [3] = (+ +).
Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities; List of volume formulas – Quantity of three-dimensional space
If one knows that the volume of a cone is (), then one can use Cavalieri's principle to derive the fact that the volume of a sphere is , where is the radius. That is done as follows: Consider a sphere of radius r {\displaystyle r} and a cylinder of radius r {\displaystyle r} and height r {\displaystyle r} .