Math 366 Lecture Notes Section 11.4 Geometry in Three Dimensions

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1 Math 366 Lecture Notes Section 11.4 Geometry in Three Dimensions Simple Closed Surfaces A simple closed surface has exactly one interior, no holes, and is hollow. A sphere is the set of all points at a given distance from a given point, the center. A sphere is a simple closed surface. A solid is a simple closed surface with all interior points. (see p. 726) 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. (see p ) 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 names 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. To draw a prism: 1) Draw one of the bases. 2) Draw vertical segments of equal length from each vertex. 3) Connect the bottom endpoints to form the second base. Use dashed segments for edges that cannot be seen. A pyramid is a polyhedron determined by a polygon and a point not in the plane of the polygon. The pyramid consists of the triangular regions determined by the point and each pair of consecutive vertices of the polygon and the polygonal region determined by 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. Pyramids are classified according to their bases. A right pyramid is one in which all the lateral faces are congruent isosceles triangles. 1

2 To draw a pyramid: 1) Draw the base. 2) Mark a point not in the plane to the base to represent the apex. 3) Draw the lateral faces by connecting vertices of the base to the apex using dashed segments for edges that cannot be seen. 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 regular polyhedron is a convex polyhedron a) whose faces are congruent regular polygonal regions, and b) the number of edges that meet at each vertex is the same for all the vertices of the polyhedron. How many regular polyhedra are there? Understand the problem Each face must be congruent to all other faces Each face is bounded by a regular polygon Find the number of different regular polyhedra Devise a plan The sum of the measures of all the angles at a vertex of a regular polyhedron is less than 360 degrees (else it would be flat at that vertex). Examine the measures of the interior angles of regular polygons to see which ones could be faces of a regular polyhedron. Carry out the plan Regular polygon Measure of interior angle Number of polygons per vertex 2

3 At least 3 figures must fit together at a vertex to make a polyhedron. If 3 angles of a regular heptagon were together, the sum of the measures of the angles would be greater than 360 degrees. What about a hexagon? The measure of an interior angle increases as the number of sides increases. Any polygon with 6 sides or more has an interior angle greater than or equal to 120 degrees, making the sum of 3 angles greater than or equal to 360 degrees. The remaining figures to consider are the equilateral triangle, square, and regular pentagon. Looking back Consider semi-regular polyhedra (soccer ball has regular pentagons and hexagons) There are 5 types of regular solid polyhedra, known as the Platonic solids (p. 729) All occur in nature (3 in crystals and 2 in skeletons of microscopic sea animals). A net is a flat pattern that can be used to construct a polyhedron. (see p ) Cylinders and Cones A cylinder is the surface formed by a line segment and a line when the segment moves so that it always remains parallel to the line, so that the endpoints trace a simple closed planar curve other than a polygon, 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. (see p. 734) 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. 3

4 A cone is the union of line segments connecting a point to each point of a simple closed curve, not coplanar with the point, the simple closed 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. (see p. 732) A right circular cone is one whose altitude 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. 4

5 Geometry Lessons for Elementary Students: Geometry Glossary: 5