BOOK VII. POLYEDRONS. DEFINITIONS. 454. A polyedron is a geometric solid bounded by planes. The intersections of the planes bounding the polyedron are called the edges; the intersections of the edges are called vertices: the portions of the planes included by the edges are called the faces. Any face may be thought of as the base. 455. Polyedrons are classified as to the number of faces required to bound them. [Question: What is the fewest number of faces necessary to form a polyedron ?] Note. The pupil will find that a few cents invested in putty or moulding clay will be well spent, since with a thinbladed knife many concrete illustrations of solids can be readily made. 456. A polyedron of four faces is called tetraedron, one of five faces a pentaedron, one of six faces a hexaedron, one of eight faces an octaedron one of ten faces a decaedron, one of twelve faces a dodecaedron, one of twenty faces an icosaedron, etc. Scholium. Having drawn on cardboard the diagrams below, cut through the heavy lines and half through the dotted lines. By folding these figures the regular polyedrons can be formed. A polyedron is convex when any plane section is a convex polygon. In the convex polyedron no face will enter the polyedron when produced. Polyedrons will be considered convex in this book unless otherwise stated. 458. A straight line joining any two vertices not in the same face is called a diagonal. 459. The volume of a solid is the number expressing its ratio. to another solid arbitrarily taken as the unit of volume. The edge of the unit is a linear unit. If a cubic cm. is contained in a given solid 50 times, its volume is 50 cubic cm. 460. Two volumes are said to be equivalent when their volumes are equal. PRISMS. 461. A prism is a polyedron two of whose faces, the bases, are equal polygons having their corresponding sides parallel, and the remaining faces are parallelograms formed by planes passing through the corresponding sides of the bases. The parallelograms are called the lateral faces. Question: What form the basal edges? the lateral edges? 462. The lateral edges of a prism are parallel and equal. Can you prove it? 463. A right section of a prism is a section formed by a plane passing at right angles to the lateral edges. 464. The altitude of a prism is the perpendicular distance between its bases. 465. Prisms are triangular, quadrangular, etc., according as their bases are triangles, quadrilaterals, etc. 466. A right prism is one whose lateral edges are perpendicular to its faces. 467. An oblique prism is one whose lateral edges are oblique to its faces. 468. A regular prism is a right prism whose bases are regular polygons. 469. The lateral faces of a prism form a prismatic surface. The faces may extend beyond the bases. |