236. THEOREM. The surface of a sphere is to the surface of a circumscribed right cylinder as two is to three. 237. THEOREM. The volume of a sphere is to the volume of a circumscribed right cylinder as two is to three. 238. THEOREM. Tangents drawn to a sphere from a common external point are equal. SECTION V. REGULAR POLYEDRONS. DEFINITIONS. 239. A polyedron has already been defined as a solid bounded by plane surfaces, the bounding planes being called faces. 240. Polyedrons are named from the number of their faces; one of four faces is called a tetraedron, one of six faces a hexaedron, etc. 241. A regular polyedron is a polyedron of which the faces are equal regular polygons, and each polyedral angle is bounded by the same number of faces. PROPOSITION XLIII. 242. THEOREM. There cannot be more than five regular polyedrons. Each face is a regular polygon (241). . And the value of one angle of a regular polygon is, in the case of the triangle 60°, But since the sum of the face angles at any vertex of a polyedral angle is less than 360° . . . (45); and since there are at least three face angles meeting at any vertex, the hexagon cannot be the face of any regular polyedron; for 120° × 3 is not less than 360°. In like manner the heptagon, octagon, etc., are excluded as faces. There remains then to be considered only the triangle, the square, and the pentagon. With respect to these, The polyedral angle may be formed by the meeting at a point of, (a) 3 triangles, since 60° x 3 = 180°, or by All other cases are excluded by the limiting value of a polyedral angle, i.e., 360°. Thus then there cannot be more than, three regular polyedrons of triangular faces; see (a), (b), (c), one regular polyedron of square faces; see (d), one regular polyedron of pentagonal faces; see (e). Five in all. Q. E. D. 243. SCHOLIUM. The five regular polyedrons are named from the number of their faces, the tetraedron, the hexaedron, the octaedron, the dodecaedron, and the icosaedron. CONSTRUCTION. REGULAR POLYEDRONS. 244. The five regular polyedrons (243) may be constructed out of cardboard in a very simple manner. Draw on the cardboard as accurately as possible the foregoing diagrams and cut them out. On the interior lines cut the cardboard about half through its thickness. The parts will then readily bend about the half-cut lines into the required form, and can be retained in place by gluing over the edges a strip of paper or linen. At E, the middle point of ABC, erect a perpendicular ED, and take the point D so that AD, which is equal to DB and to DC, shall be equal to AB. |