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Polyhedra

A polyhedron is a closed, three-dimensional figure formed by joining polygons at their edges. The individual polygons are called faces. The edges of the polygons are still called edges and the vertices of the polygons are still called vertices. Several examples of polyhedra are illustrated below.

pyramid 1 prism 1 polyhedron 1

Euler's Formula

Let P be a convex polyhedron. If P has V vertices, E edges and F faces then V - E + F = 2. You can confirm this using the polyhedra above as examples.


Prisms

A prism is a polyhedron were two of the sides are parallel, congruent polygons and the remaining sides are parallelograms. The parallel, congruent sides are called bases. (The middle figure above is an example.) An altitude of a prism is a segment that goes from one base to the plane of the second and is perpendicular to both. The lateral surface area or lateral area of a prism is the total area of the side faces, i.e. all of the faces except the bases. A right prism is a prism where the lateral faces are rectangles.

a prism a right prism a prism
a prism a right prism a prism w/ altitudes in red

Notice how, in the third example, altitudes don't have to be inside the prism. This is why the definition specifies that an altitude only has to go to "the plane of the second base" rather than to the base itself.

Note that all of the following formulae only apply to right prisms.

Formulae

Lateral Area If p is the perimeter of the base of a right prism and h is the length of an altitude (i.e. the height) then the lateral surface area LA is LA = ph.
Total Area If A is the total area of the prism, LA is the lateral area and BA is the area of a base then A = LA + 2BA.
Volume If A is the area of the base of a right prism and h is the length of an altitude then the volume of the prism is V = Ah.

Pyramids

A pyramid is a polyhedron whose base is a polygon and whose lateral faces are triangles that share a common point called the vertex. An altitude of a pyramid is a segment that goes from the vertex to the plane of the base and is perpendicular to the base. The lateral surface area or lateral area of a pyramid is the total area of the side faces, i.e. all of the faces except the base. A regular pyramid is a pyramid where the base is a regular polygon, the lateral faces are isosceles triangles and the altitude meets the base at its center. The length of the altitudes of the faces in a regular pyramid is called the slant height.

a pyramid a regular pyramid a pyramid
a pyramid a regular pyramid a pyramid w/ altitude in red

Notice how, in the third example, the altitude isn't inside the prism. This is why the definition specifies that an altitude only has to go to "the plane of the base" rather than to the base itself.

Note that all of the following formulae only apply to regular pyramids.

Formulae

Lateral Area If l is the slant height of a regular pyramid, b is the length of a side of the pyramid’s base and n is the number of sides then the pyramid’s lateral area is LA = bln / 2. If p is the perimeter of the base then LA = lp / 2.
Volume For any regular pyramid, if V is the volume, B is the area of the base and h is the height then V = Bh / 3.