Definitions of Cones and Cylinders
Date: 03/02/2004 at 12:37:59 From: Ms. Hamilton Subject: Grade Five Geometry Are cones and cylinders pyramids, prisms or neither? My class cannot reach an agreement on this subject!
Date: 03/02/2004 at 13:08:45 From: Doctor Peterson Subject: Re: Grade Five Geometry Hi, Ms. Hamilton (and class!) The only way to decide is to look up the definitions and see if they fit; the trouble is that you'll find slightly different definitions here and there. And elementary texts aren't known for giving mathematically precise, and fully general, definitions, so that might not be your best source. One source of (advanced) definitions is Eric Weisstein's World of Mathematics http://mathworld.wolfram.com/ which is listed in our FAQ. You might like going through that with the class. As a sample, here are his definitions of "cylinder" and "prism": In common usage, the term "cylinder" refers to a solid of circular cross section in which the centers of the circles all lie on a single line (i.e., a right circular cylinder). In mathematical usage, "cylinder" is commonly taken to refer to only the lateral sides of this solid, excluding the top and bottom caps. An oblique prism is a polyhedron with two congruent polygonal faces and all remaining faces parallelograms (left figure). A right prism is a prism in which the top and bottom polygons lie on top of each other so that the vertical polygons connecting their sides are not only parallelograms, but rectangles (right figure). If, in addition, the upper and lower bases are rectangles, then the prism is known as a cuboid. Unfortunately, he doesn't give a good definition of the cylinder in the most general sense. Note that a cylinder in the common, specialized sense is really a "right circular cylinder". It is called "circular" because its base is circular; more generally, you can make a cylinder from ANY closed plane curve, not just a circle. And if that "curve" is a polygon (to a mathematician, there's nothing wrong with a "curve" having straight sides or corners), then you have a prism. So a prism is actually a polygonal cylinder. Likewise, a pyramid is a polygonal cone. But a (circular) cylinder is not a prism, and a (circular) cone is not a pyramid, because the prism and the pyramid must be polyhedra, with flat sides. You can see definitions that fit what I just described in our FAQ: http://mathforum.org/dr.math/faq/formulas/faq.cylinder.html A cylinder is a surface generated by a family of all lines parallel to a given line (the generatrix) and passing through a curve in a plane (the directrix). A right section is the curve formed by the intersection of the surface and a plane perpendicular to the generatrix. The parallel bases of a cylinder may form any angle with the axis. A cone is a surface generated by a family of all lines through a given point (the vertex) and passing through a curve in a plane (the directrix). More commonly, a cone includes the solid enclosed by a cone and the plane of the directrix. The region of the plane enclosed by the directrix is called a base of the cone. The perpendicular distance from the vertex to the plane of the base is the height of the cone. To interpret this definition, see Definition For Cylinder without Big Words http://mathforum.org/library/drmath/view/55052.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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