BOOK VII.

POLYEDRONS.

DEFINITIONS.

461. A polyedron is a solid bounded by polygons.

The bounding polygons are called the faces of the polyedron; their sides are called the edges, and their vertices the vertices.

A diagonal of a polyedron is a straight line joining any two vertices not in the same face.

462. The least number of planes which can form a polyedral angle is three.

Whence, the least number of polygons which can bound a polyedron is four.

A polyedron of four faces is called a tetraedron; of six faces, a hexaedron; of eight faces, an octaedron; of twelve faces, a dodecaedron; of twenty faces, an icosaedron.

463. A polyedron is called convex when the section made by any plane is a convex polygon (§ 121).

All polyedrons considered hereafter will be understood to be convex.

464. The volume of a solid is its ratio to another solid, called the unit of volume, adopted arbitrarily as the unit of measure (§ 180).

The usual unit of volume is a cube (§ 474) whose edge is some linear unit; for example, a cubic inch or a cubic foot.

465. Two solids are said to be equivalent when their volumes are equal.

PRISMS AND PARALLELOPIPEDS.

DEFINITIONS.

466. A prism is a polyedron, two of whose faces are equal polygons lying in parallel planes, having their homologous sides parallel, the other faces being parallelograms (§ 110). The equal and parallel faces are called the bases of the prism, and the other faces the lateral faces; the edges which are not

sides of the bases are called the lateral edges, and the sum of the areas of the lateral faces the lateral area.

The altitude is the perpendicular distance between the planes of the bases.

467. The following is given for convenience of reference: The bases of a prism are equal.

468. It follows from the definition of § 466 that the lateral edges of a prism are equal and parallel.

(§ 106, I)

469. A prism is called triangular, quadrangular, etc., according as its base is a triangle, quadrilateral, etc.

470. A right prism is a prism whose lat

eral edges are perpendicular to its bases. The lateral faces are rectangles (§ 398).

An oblique prism is a prism whose lateral

edges are not perpendicular to its bases.

471. A regular prism is a right prism whose base is a regular polygon.

472. A truncated prism is a portion of a prism included between the base, and a plane, not parallel to the base, cutting all the lateral edges.

The base of the prism and the section made by the plane are called the bases of the truncated prism.

473. A right section of a prism is a section made by a plane cutting all the lateral edges, and perpendicular to them.

474. A parallelopiped is a prism whose bases are parallelograms; that is, all the faces are parallelograms.

A right parallelopiped is a parallelopiped whose lateral edges are perpendicular to its bases.

A rectangular parallelopiped is a right parallelopiped whose bases are rectangles; that is, all the faces are rectangles.

A cube is a rectangular parallelopiped whose six faces are all squares.

PROP. I. THEOREM.

475. The sections of a prism made by two parallel planes which cut all the lateral edges, are equal polygons.

Given planes CF and C'F' cutting all the lateral edges of prism AB.

To Prove section CDEFG:

Proof. We have CD | C'D', DE D'E', etc.

=

section CD'E'F'G'.

(§ 414)

(§ 107)

Also CDE <C'D'E', / DEF = D'E'F", etc. (§ 426)

Then, polygons CDEFG and C'D'E'F'G', being mutually equilateral and mutually equiangular, are equal.

(§ 124)

476. Cor.

The section of a prism made by a plane paral

lel to the base is equal to the base.

PROP. II. THEOREM.

477. Two prisms are equal when the faces including a triedral angle of one are equal respectively to the faces including a triedral angle of the other, and similarly placed.

Given, in prisms AH and A'H', faces ABCDE, AG, and AL equal respectively to faces A'B'C'D'E', A'G', and A'L'; the equal parts being similarly placed.

To Prove prism AH prism A'H'.

=

Proof. We have EAB, EAF, and FAB equal respectively to E'A'B', E'A'F', and F'A'B'.

(§ 66)

.. triedral A-BEF= triedral A'-B'E'F". (§ 460, 1) Then, prism A'H' may be applied to prism AH in such a way that vertices A', B', C', D', E', G', F", and L' shall fall at A, B, C, D, E, G, F, and L, respectively.

Now since the lateral edges of the prisms are , edge C'H' will fall on CH, D'K' on DK, etc.

(§ 53) And since points G', F", and L' fall at G, F, and L, respectively, planes LH and L'H' coincide. (§ 395, II) Then points H' and K' fall at H and K, respectively. Hence, the prisms coincide throughout, and are equal.

478. Cor. Two right prisms are equal when they have equal bases and equal altitudes; for by inverting one of the prisms if necessary, the equal faces will be similarly placed.

479. Sch. The demonstration of § 477 applies without change to the case of two truncated prisms.

480. An oblique prism is equivalent to a right prism, having for its base a right section of the oblique prism, and for its altitude a lateral edge of the oblique prism.

Given FK' a right prism, having for its base FK a right section of oblique prism AD', and its altitude FF" equal to AA', a lateral edge of AD'.

Proof. In truncated prisms AK and A'K', faces FGHKL and F'G'H'K'L' are equal. (§ 475) Therefore, A'K' may be applied to AK so that vertices. F", G', etc., shall fall at F, G, etc., respectively.

Then, edges A'F", B'G', etc., will coincide in direction with AF, BG, etc., respectively.

C'H', etc.

(§ 399)

But since, by hyp., FF' = AA', we have AF = A'F'. In like manner, BG = B'G', CH = = Hence, vertices A', B', etc., will fall at A, B, etc., respectively.

Then, A'K' and AK coincide throughout, and are equal. Now taking from the entire solid AK' truncated prism A'K', there remains prism AD'.

And taking its equal AK, there remains prism FK'.