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Fig. 17-4
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Fig. 17-5
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A cubic unit is a cube whose edge measures 1 unit. Thus, a cubic inch is a cube whose edge is 1 in long. A parallelepiped is a prism bounded by six parallelograms (Fig. 17-5). Hence, the rectangular solid and cube are special parallelepipeds. The table that follows shows some relationships among polygons in plane geometry and the corresponding relationships among polyhedra in solid geometry. Plane Geometry Solid Geometry Polyhedron Polygon Quadrilateral Prism Parallelogram Parallelepiped Rectangle Square Rectangular solid Cube Pyramids A pyramid is a polyhedron whose base is a polygon and whose other faces meet at a point, its vertex. The base (B in Fig. 17-6) may have any number of sides. However, the other faces must be triangles. The distance from the vertex to the base is equal to the measure of the altitude or height h, a line from the vertex at a right angle to the base.
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CHAPTER 17 Extending Plane Geometry into Solid Geometry
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B Regular Pyramid (square base)
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A regular pyramid is a pyramid whose base is a regular polygon and whose altitude joins the vertex and the center of the base. A frustum of a pyramid is the part of a pyramid that remains if the top of the pyramid is cut off by a plane parallel to the base. Note in Fig. 17-6 that its lateral faces are trapezoids. Cones A circular cone (Fig. 17-7) is a solid whose base is a circle and whose lateral surface comes to a point. (A circular cone is usually referred to simply as a cone.)
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B Cone
Right Circular Cone
Frustum of a Cone
Fig. 17-7
A right circular cone is formed by revolving a right triangle about one of its legs. This leg becomes the altitude h of the cone, and the other becomes the radius r of the base. A frustum of a cone is the part of a cone that remains if the top of the cone is cut off by a plane parallel to the base. Cylinders A circular cylinder (Fig. 17-8) is a solid whose bases are parallel circles and whose cross-sections parallel to the bases are also circles. (A circular cylinder is usually referred to simply as a cylinder.) A right circular cylinder is a circular cylinder such that the line joining the centers of the two bases is perpendicular to the radii of these bases. The line joining the centers is the height h of the cylinder, and the radius of the bases is the radius r of the cylinder.
CHAPTER 17 Extending Plane Geometry into Solid Geometry
B B Cylinder
Cylinder of Revolution
Fig. 17-8
Spheres A sphere is a solid such that every point on its surface is at an equal distance from the same point, its center. A sphere is formed by revolving a semicircle about its diameter as an axis. The outer end of the radius perpendicular to the axis generates a great circle, while the outer ends of other chords perpendicular to the diameter generate small circles. Thus, sphere O in Fig. 17-9 is formed by the rotation of semicircle ACB about AB as an axis. In the process, point C generates a great circle while points E and G generate small circles.
N (North Pole)
P R I M E M E RIDIAN
TIC AN ATL EAN OC
EQUATOR
B S (South Pole) Fig. 17-10
Fig. 17-9
The way in which points on the earth s surface are located is better understood if we think of the earth as a sphere formed by rotating semicircle NQS in Fig. 17-10, which runs through Greenwich, England (near London), about NS as an axis. Point O, halfway between N and S, generates the equator, EQ. Points A and B generate parallels of latitude, which are small circles on the earth s surface parallel to the equator. Each position of the rotating semicircle is a semi-meridian, or longitude. (A meridian is a great circle passing through the North and South Poles.) The meridian through Greenwich is called the Prime Meridian. Using the intersection of the equator and the Prime Meridian as the origin, we would locate New York City r r at 40 481 North Latitude and 73 571 West Longitude. The heavy arc shown on the globe is an arc of a great 2 2 circle through New York City and London. Such an arc is the shortest distance between two points on the earth s surface. We could find this line by stretching a rubber band tightly between New York City and London on a globe.
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