Download Section 9.4 - Geometry in Three Dimensions

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Kendra
Kilmer May 25, 2008
Section 9.4 - Geometry in Three Dimensions
Simple Closed Surfaces:
• A simple closed surface has exactly one interior, has no holes, and is hollow. It separates space into
interior, surface, and exterior.
• A sphere is the set of all points at a given distance from a given point, the center.
• A solid is the set of all points on a simple closed surface along with all its interior points.
• A polyhedron is a simple closed surface made up of polygonal regions, or faces. The vertices of the
polygonal regions are the vertices of the polyhedron, and the sides of each polygonal region are the
edges of the polyhedron.
• A prism is a polyhedron in which two congruent faces lie in parallel planes and the other faces are
bounded by parallelograms. The parallel faces of a prism are the bases of the prism. A prism is usually
named after its bases. The faces other than the bases are the lateral faces of a prism.
• A right prism is one in which the lateral faces are all bounded by rectangles.
• An oblique prism is one in which some of the lateral faces are not bounded by rectangles.
• A pyramid is a polyhedron determined by a polygon and a point not in the plane of the polygon. The
polygonal region is the base of the pyramid, and the point is the apex. The faces other than the base are
lateral faces.
• A pyramid is a right pyramid if all its lateral faces are congruent isosceles triangels.
Example 1: Draw a right pentagonal prism.
Example 2: Draw a pentagonal pyramid.
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c
Kendra
Kilmer May 25, 2008
Regular Polyhedra:
• A polyhedron is a convex polyhedron if and only if the segment connecting any two points in the
interior of the polyhedron is itself in the interior.
• A concave polyhedron is a polyhedron that is not convex.
• A regular polyhedron is a convex polyhedron whose faces are congruent regular polygonal regions
such that the number of edges that meet at each vertex is the same for all the vertices of the polyhedron.
Example 3: How many regular polyhedra are there?
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c
Kendra
Kilmer May 25, 2008
Cylinders and Cones:
• A cylinder is the surface formed by moving a segment (keeping it parallel to the original segment) to
form a simple closed non-polygonal curve at its ends, along with the simple closed curves, and their
interiors. The simple closed curves traced by the endpoints of the segment, along with their interiors,
are the bases of the cylinder; the remaining points constitute the lateral surface of the cylinder.
• A circular cylinder is one whose base is a circular region.
• A right cylinder is one in which the line segment forming the cylinder is perpendicular to a base.
• An oblique cylinder is any cylinder that is not a right cylinder.
• A cone is the union of line segments connecting a point not on a simple closed non-polygonal curve to
each point of the simple closed non-polygonal curve, along with the simple closed non-polygonal curve
and the interior of the curve. The point is the vertex of the cone; the points not in the base constitute the
lateral surface of the cone. A line segment from the vertex perpendicular to the plane of the base is the
altitude of the cone.
• A right circular cone is one whose altitued intersects the circular base at the center of the circle.
• An oblique circular cone is a circular cone that is not a right circular cone.
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