BOOK VIII. POLYEDRONS. DEFINITIONS. 435. A POLYEDRON is a solid, or volume, bounded by planes. The bounding planes are called the faces of the polyedron; and the lines of intersection of the faces are called the edges of the polyedron. 436. A PRISM is a polyedron having two of its faces equal and parallel polygons, and the other faces parallelo grams. The equal and parallel polygons are called the bases of the prism, and the parallelograms its lateral faces. The lateral faces taken together constitute the lateral or convex surface of the prism. Thus the polyedron ABCDE-K is a prism, having for its bases the equal and parallel polygons ABCDE, FGHIK, and for its lateral faces the parallelograms ABGF, BCH G, &c. The principal edges of a prism are those which join the corresponding angles of the bases; as A F, B G, &c. 437. The altitude of a prism is a perpendicular drawn from any point in one base to the plane of the other. 438. A RIGHT PRISM is one whose principal edges are perpendicular to the planes of its bases. Each of the edges is then equal to the altitude of the prism. Every other prism is oblique, and has each edge greater than the altitude. 439. A prism is triangular, quadrangular, pentangular, hexangular, &c., according as its base is a triangle, a quadrilateral, a pentagon, a hexagon, &c. 440. A PARALLELOPIPEDON is a prism whose bases are parallelograms; as the prism A B C D - H. The parallelopipedon is rectangular when all its faces are rectangles; as the parallelopipedon ABCD-H. A E 441. A CUBE, or REGULAR HEXAEDRON, is a rectangular parallelopipedon having all its faces equal squares; as the paral- A lelopipedon A B C D - H. A B H S Η 442. A PYRAMID is a polyedron of which one of the faces is any polygon, and all the others are triangles meeting at a common point. The polygon is called the base of the pyramid, the triangles its lateral faces, and the point at which the triangles meet its vertex. The lateral faces taken together constitute the lateral or convex surface of the pyramid. B Thus the polyedron A B C D E - S is a pyramid, having for its base the polygon A B C D E, for its lateral faces the triangles A S B, BSC, CSD, &c., and for its vertex the point S. G G C 443. The ALTITUDE of a pyramid is a perpendicular drawn from the vertex to the plane of the base. 444. A pyramid is triangular, quadrangular, &c., according as its base is a triangle, a quadrilateral, &c. 445. A RIGHT PYRAMID is one whose base is a regular polygon, and the perpendicular drawn from the vertex to the base passes through the centre of the base. In this case the perpendicular is called the axis of the pyramid. 446. The SLANT HEIGHT of a right pyramid is a line drawn from the vertex to the middle of one of the sides of the base. 447. A FRUSTUM of a pyramid is the part of the pyramid included between the base and a plane cutting the pyramid parallel to the base. 448. The ALTITUDE of the frustum of a pyramid is the perpendicular distance between its parallel bases. 449. The SLANT HEIGHT of a frustum of a right pyramid is that part of the slant height of the pyramid is intercepted between the bases of the frustum. 450. The Axis of the frustum of a pyramid is that part of the axis of the pyramid which is intercepted between the bases of the frustum. 451. The DIAGONAL of a polyedron is a line joining the vertices of any two of its angles which are not in the same face. 452. SIMILAR POLYEDRONS are those which are bounded by the same number of similar faces, and have their polyedral angles respectively equal. 453. A REGULAR POLYEDRON is one whose faces are all equal and regular polygons, and whose polyedral angles are all equal to each other. 454. The convex surface of a right prism is equal to the perimeter of its base multiplied by its altitude. Let ABCDE-K be a right prism; then will its convex surface be equal to the perimeter of its base, AB+BC+CD+DE+EA, multiplied by its altitude A F. B C For, the convex surface of the prism is equal to the sum of the parallelograms AG, BH, CI, DK, EF (Art. 436). Now, the area of each of those parallelograms is equal to its base, AB, BC, CD, &c., multiplied by its altitude, AF, BG, CH, &c. (Prop. V. Bk. IV.). But the altitudes AF, BG, CH, &c. are each equal to A F, the altitude of the prism. Hence, the area of these parallelograms, or the convex surface of the prism, is equal to A F G K H (AB+ BC + CD+DE+E A) × AF; or the product of the perimeter of the prism by its altitude. 455. Cor. If two right prisms have the same altitude, their convex surfaces are to each other as the perimeters of their bases. PROPOSITION II. - THEOREM. 456. In every prism, the sections formed by parallel planes are equal polygons. Let the prism ABCDE-K be intersected by the parallel planes NP, SV; then are the sections NOPQR, STVXY equal polygons. For the sides ST, NO are parallel, being the intersections of two parallel planes with a third plane ABG F (Prop. XIII. Bk. VII.); these same sides ST, NO, are included between the parallels NS, OT, which are sides of the prism; hence NO is equal to ST. For like reasons, the sides OP, PQ, QR, &c. of the section NOPQR, are respectively equal to the sides TV, VX, XY, &c. of the section STVXY; and since the equal sides are at the same time parallel, it follows that the angles NOP, OPQ, &c. of the first section are respectively equal to the angles STV, TVX of the second (Prop. XVI. Bk. VII.). Hence, the two sections NOPQR, STVXY, are equal polygons. B A F G S E N K H K R T E O 457. Cor. Every section made in a prism parallel to its base, is equal to that base. L PROPOSITION III.-THEOREM. 458. Two prisms are equal, when the three faces which form a triedral angle in the one are equal to those which form a triedral angle in the other, each to each, and are similarly situated. R Let the two prisms ABCDE-K and LMOPQ-Y have the faces which form the triedral angle B equal to the faces which form the triedral angle M; that is, the base ABCDE equal to the base LMNOPQ, the parallelogram ABG F equal to the parallelogram LMSR, and the parallelogram HG equal to MOTS; then the two prisms are equal. B C M O X Q S I Y प H |