BOOK VII.

POLYHEDRONS, CYLINDERS, AND CONES.

POLYHEDRONS.

584. DEF. A polyhedron is a solid bounded by planes.

The bounding planes are called the faces, the intersections" of the faces, the edges, and the intersections of the edges, the vertices, of the polyhedron.

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

586. DEF. A section of a polyhedron is the figure formed by its intersection with a plane passing through it.

587. DEF. A polyhedron is convex if every section is a convex polygon.

Only convex polyhedrons are considered in this work.

588. DEF. A polyhedron of four faces is called a tetrahedron; one of six faces, a hexahedron; one of eight faces, an octahedron; one of twelve faces, a dodecahedron; one of twenty faces, an icosahedron.

NOTE.

Full lines in the figures of solids represent visible lines, dashed lines represent invisible lines.

PRISMS AND PARALLELOPIPEDS.

589. DEF. A prism is a polyhedron of which two faces are equal polygons in parallel planes, and the other faces are parallelograms.

The equal polygons are called the bases of the prism, the parallelograms, the lateral faces, and the intersections of the lateral faces, the lateral edges of the prism.

The sum of the areas of the lateral faces of a prism is called its lateral area.

Right Prism.

Prism.

590. DEF. The altitude of a prism is the perpendicular distance between the planes of its bases.

591. DEF. A right prism is a prism whose lateral edges are perpendicular to its bases.

592. DEF. A regular prism is a right prism whose bases are regular polygons.

593. DEF. An oblique prism is a prism whose lateral edges are oblique to its bases.

594. The lateral edges of a prism are equal. The lateral edges of a right prism are equal to the altitude.

595. DEF. Prisms are called triangular, quadrangular, etc., according as their bases are triangles, quadrilaterals, etc.

Triangular Prism.

596. DEF. A parallelopiped is a prism whose bases are parallelograms.

597. DEF. A right parallelopiped is a parallelopiped whose lateral edges are perpendicular to the bases.

part of a prism included between the base and a section made

by a plane oblique to the base.

PROPOSITION I. THEOREM.

605. The sections of a prism made by parallel planes cutting all the lateral edges are equal polygons.

Let the prism AD be intersected by parallel planes cutting all the lateral edges, making the sections GK, G'K'.

Το prove that

GK = G'K'.

Proof. The sides GH, HI, IK, etc., are parallel, respectively, to the sides G'H', H'I', I'K', etc.

$ 528 The sides GH, HI, IK, etc., are equal, respectively, to G'H', H'I', I'K', etc.

§ 180 The GHI, HIK, etc., are equal, respectively, to & G'H'I', H'I'K', etc.

$ 534

§ 203

Q. E. D.

606. COR. Every section of a prism made by a plane parallel to the base is equal to the base; and all right sections of a prism are equal.

Ex. 630. The diagonals of a parallelopiped bisect one another.

Ex. 631. The lateral faces of a right prism are rectangles.

Ex. 632. Every section of a prism made by a plane parallel to the lateral edges is a parallelogram.

607. The lateral area of a prism is equal to the product of a lateral edge by the perimeter of the right section.

Let GHIKL be a right section of the prism AD', S its lateral area,

E a lateral edge, and P the perimeter of the right section.

the area of BC' CC' × HI = EX HI, and so on.

Therefore, S, the sum of these parallelograms, is equal to

608. COR. The lateral area of a right prism is equal to the product of the altitude by the perimeter of the base.

Ex. 633. Find the lateral area of a right prism, if its altitude is 18 inches and the perimeter of its base 29 inches.