How are some common solidsdefined?
There are numerous objects defined in solid geometry. The following lists the most
common (and some interesting) ones:
Cone—A cone can be both a surface and a solid. A solid cone is bounded by a
region enclosed in a closed curve on a plane and a surface formed by segments joining
each point of the closed curve to a point that is not in the plane. (Note: Two solid
cones seen with their pointed ends together help define conic sections; for more about
conic sections, see elsewhere in this chapter.)
Pyramid—A pyramid is a polyhedron (see above) with one face a polygon and all
other faces as triangles with a common endpoint (vertex). They are named based on
the polygon’s base, such as the triangular pyramid, square pyramid, and so on. Some
of the most famous “solid pyramids” are the sandstone pyramids of Egypt. These are
actually called right square pyramids because the base of the polygon is a square and
the vertical line from the vertex meets the center of the base.
Cylinder—A cylinder can be both a surface and a solid. A solid cylinder is one that
forms by rotating a circle about an axis through the midpoints of the opposite side; it
is also called a right circular cylinder. One of the most well-known cylinders is possi-
bly sitting right next to you: A coffee cup, with its cylindrical shape, and the bases (in
186 this case, the base and rim) considered to be congruent circles.
What are the Platonic solids?
P
latonic solids are also called regular solids, regular figures (a term also used
in reference to polygons), regular polyhedra, or “cosmic figures.” These
solids are convex polyhedra that have equal faces made up of congruent convex
regular polygons. (To compare in terms of a two-dimensional polygon, a regular
figure means that both the sides and the angles between them are equal.)There are considered to only be five of these solids: the cube, dodecahedron,
icosahedron, octahedron, and tetrahedron. These solids were described by Greek
philosopher Plato (c. 428–348 BCE) in his work Timaeus—thus the name Platon-
ic solids. His definitions were certainly more whimsical than today’s, as he
believed the major “elements” were made up of atoms shaped like certain poly-
hedra. He associated the tetrahedron with the “element” fire, the cube with the
earth, the icosahedron with water, the octahedron with air, and the dodecahe-
dron with the material that makes up the constellations and heavens. The math-
ematical proofs of these solids were worked out long ago by Greek mathemati-
cian and geometrician Euclid (c. 325–c. 270 BCE) in the last part of his Elements
(for more information about Euclid, see elsewhere in this chapter, as well as
“Foundations of Mathematics”).