BOOK VII.

POLYEDRONS.

DEFINITIONS.

1. A POLYEDRON is a volume bounded by polygons. The bounding polygons are called faces of the polyedron; the lines in which the faces meet, are called edges of the polyedron; the points in which the edges meet, are called vertices of the polyedron.

2. A PRISM is a polyedron in which two of the faces are polygons equal in all respects, and having their homologous sides parallel. The other faces are parallelograms (B. I., P. XXX.).

The equal polygons are called bases of the prism; one the upper, and the other

the lower base; the parallelograms taken together make up the lateral or convex surface of the prism; the lines in which the lateral faces meet, are called lateral edges, and the lines in which the lateral faces meet either base are called basal edges of the prism.

3. The ALTITUDE of a prism is the perpendicular distance between the planes of its bases.

4. A RIGHT PRISM is one whose lateral edges are perpendicular to the planes of the

bases.

In this case, any lateral edge is equal to the altitude.

GEOMETRY.

5. An OBLIQUE PRISM is one whose lateral edges are oblique to the planes of the bases.

In this case, any lateral edge is greater than the altitude.

6. Prisms are named from the number of sides of their bases; a triangular prism is one whose bases are triangles; a pentagonal prism is one whose bases are pentagons, &c.

7. A PARALLELOPIPEDON is a prism whose bases are parallelograms.

A Right Parallelopipedon is one whose lateral edges are perpendicular to the planes of the bases.

A Rectangular Parallelopipedon is one whose faces are all rectangles.

A Cube is a rectangular parallelopipedon whose faces are squares.

8. A PYRAMID is a polyedron bounded by a polygon called the base, and by triangles meeting at a common point, called the vertex of the pyramid.

The triangles taken together make up the lateral or convex surface of the pyramid ; the lines in which the lateral faces meet, are called the lateral edges, and the lines in which the lateral faces meet the base are called basal edges of the pyramid.

9. Pyramids are named from the number of sides of their bases; a triangular pyramid is one whose base is a triangle; a quadrangular pyramid is one whose base is a quadrilateral, and so on.

10. The ALTITUDE of a pyramid is the perpendicular distance from the vertex to the plane of its base,

11. A RIGHT PYRAMID is one whose base is a regular polygon, and in which the perpendicular, drawn from the vertex to the plane of the base, passes through the centre of the base.

This perpendicular is called the axis of the pyramid.

12. The SLANT HEIGHT of a right pyramid, is the perpendicular distance from the vertex to any side of the base.

13. A TRUNCATED PYRAMID is is that portion of a pyramid included between the base and any plane which cuts the pyramid.

When the cutting plane is parallel to the base, the truncated pyramid is called a FRUSTUM OF A PYRAMID, and the inter

section of the cutting plane with the pyramid, is called the upper base of the frustum; the base of the pyramid is called the lower base of the frustum.

14. The ALTITUDE of a frustum of a pyramid, is the perpendicular distance between the planes of its bases.

15. The SLANT HEIGHT of a frustum of a right pyramid, is that portion of the slant height of the pyramid which lies between the planes of its upper and lower bases.

16. SIMILAR POLYEDRONS are those which are bounded by the same number of similar polygons, similarly placed. Parts which are similarly placed, whether faces, edges, or angles, are called homologous.

17. A DIAGONAL of a polyedron, is a straight line joining the vertices of two polyedral angles not in the same

face.

18. The VOLUME OF A POLYEDRON is its numerical value expressed in terms of some other polyedron taken as a unit.

The unit generally employed is a cube constructed on the linear unit as an edge.

PROPOSITION I. THEOREM.

The convex surface of a right prism is equal to the perimeter of either base multiplied by the altitude.

Let ABCDE-K be a right prism: then is its convex surface equal to,

(AB+ BC + CD + DE + EA) × AF.

For, the convex surface is equal to the sum of all the rectangles AG, BH, CI, DK, EF, which compose it. Now, the altitude of each of the rectangles AF, BG, CH, &c., is equal to the altitude of the prism, and the area of each rectangle is equal to its base multiplied by its altitude (B. IV., P. V.): hence, the sum of these rectangles, or the convex surface of the prism, is equal to,

(AB+BC+CD+DE+EA) x AF;

K

F

D

that is, to the perimeter of the base multiplied by the altitude; which was to be proved.

Cor. If two right prisms have the same altitude, their convex surfaces are to each other as the perimeters of their bases.

In any prism, the sections made by parallel planes are polygons equal in all respects.

Let the prism AH be intersected by the parallel planes NP, SV: then are the sections NOPQR, STVXY, equal polygons.

N

E

Y

K

Ꭴ

Q

Х

H

For, the sides NO, ST, are parallel, being the intersections of parallel planes with a third plane ABGF; these sides, NO, ST, are included between the parallels NS, OT: hence, NO is equal to ST (B. I., P. XXVIII., C. 2). For like reasons, the sides OP, PQ, QR, &c., of NOPQR, are equal to the sides TV, VX, &c., of STVXY, each to each; and since the equal sides are parallel, each to each, it follows that the angles NOP, OPQ, &c., of the first section, are equal to the angles STV, TVX, &c., of the second section, each to each (B. VI., P. XIII.): hence, the two sections NOPQR, STVXY, are equal in all respects; which was to be proved.

B

Cor. The bases of a prism and any section of a prism parallel to the bases, are equal in all respects.

If a pyramid is cut by a plane parallel to the base: 1. The edges and the altitude are divided proportionally: 2o. The section is a polygon similar to the base.

Let the pyramid S-ABCDE, whose altitude is SO, be cut by the plane abcde, parallel to the base ABCDE.