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Handbook of mathematics for engineers and scienteists part 15

This point is equidistant from all sides and is called the mc^rt/^r (the center of the inclrcle of the triangle). Concerning the radius r of the incircle. see Paragraph 3.1.1-3. The angle bisector through a vertex cuts the opposite side in ratio proportional to the adjacent sides of the triangle. | 66 Elementary Geometry TABLE 3.4 Regular polyhedra a is the edge length No. Name Number of faces and its shape Number of vertices Number of edges Total surface area Volume 1 2 Tetrahedron 4 triangles 6 squares 4 8 6 12 a2 73 6a2 a3 72 Cube 12 a3 3 Octahedron 8 triangles 6 12 2a2 73 a3 72 3 y 15 7V5 4 Dodecahedron 12 pentagons 20 30 3a2 25 10V5 5 Icosahedron 20 triangles 12 30 3a2 V3 5 3 75 a b H Figure 3.32. A cylindrical surface a . A cylinder b . the generator then the lateral surface area Slat and the volume V of the cylinder are given by the formulas Slat PH Psec l V- S H-S l 3.2.3.1 V SbasH Ssed . In a right cylinder the bases are perpendicular to the generator. In particular if the bases are disks then one speaks of a right circular cylinder. The volume the lateral surface area and the total surface area of a right circular cylinder are given by the formulas V nR2H Slat 2nRH S 2nR R H 3.2.3.2 where R is the radius of the base. A right circular cylinder is also called a round cylinder or simply a cylinder. 2 . The part of a cylinder cut by a plane nonparallel to the base is called a truncated cylinder Fig. 3.33a . The volume the lateral surface area and the total surface area of a truncated cylinder 3.2. Solid Geometry 67 a c b Figure 3.33. A truncated cylinder a a hoof b and a cylindrical tube c . are given by the formulas V R H H 2 Slat nRH H 3.2.3.3 S nR H1 H2 R H2 - Hi 2 2 where Hi and H2 are the maximal and minimal generators. 3 . A segment of a round cylinder a hoof is a portion of the cylinder cut by a plane that is nonparallel to the base and intersects it. If R is the radius of the cylindrical segment h is the height of the hoof and b is its width for the other notation see Fig. 3.33b then the volume V and the lateral surface area Slat of the hoof can be determined by the formulas 3 3 V 3R2 - a2 3R2 b - R a sin a-------------3-----a cos a Slat 7 b R a aL b 3.2.3.4 where a 2p is measured in radians. 4 . A solid bounded by two closed cylindrical .