http://www.ck12.org Chapter 11. Surface Area and Volume
11.1 Polyhedrons
Here you’ll learn how to identify polyhedron and regular polyhedron and the connections between the numbers of
faces, edges, and vertices in polyhedron.
What if you were given a solid three-dimensional figure, like a carton of ice cream? How could you determine how
the faces, vertices, and edges of that figure are related? After completing this Concept, you’ll be able to use Euler’s
Theorem to answer that question.
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Guidance
Apolyhedronis a 3-dimensional figure that is formed by polygons that enclose a region in space. Each polygon
in a polyhedron is called aface. The line segment where two faces intersect is called anedgeand the point of
intersection of two edges is avertex. There are no gaps between the edges or vertices in a polyhedron. Examples
of polyhedrons include a cube, prism, or pyramid. Non-polyhedrons are cones, spheres, and cylinders because they
have sides that are not polygons.
Aprismis a polyhedron with two congruent bases, in parallel planes, and the lateral sides are rectangles. Prisms
are explored in further detail in another Concept.
Apyramidis a polyhedron with one base and all the lateral sides meet at a common vertex. The lateral sides are
triangles. Pyramids are explored in further detail in another Concept.
All prisms and pyramids are named by their bases. So, the first prism would be a triangular prism and the second
would be an octagonal prism. The first pyramid would be a hexagonal pyramid and the second would be a square
pyramid. The lateral faces of a pyramid are always triangles.
Euler’s Theoremstates that the number of faces(F), vertices(V), and edges(E)of a polyhedron can be related
such thatF+V=E+2.