862. COROLLARY. The volume of the solid generated by the revolution of any triangle about one of its sides as axis, is one-third the product of the triangle's area into the length of the circle described by its vertex.

V = Zπrs.

PRISMATOID.

863. A Prismatoid is a polyhedron whose bases are any two polygons in parallel planes, and whose lateral faces are triangles

determined by so joining the vertices of these bases that each lateral edge, with the preceding, forms a triangle with one side of either base.

864. A number of different prismatoids thus pertain to the same two bases.

865. If two basal ed es which form with the same lateral edge two sides of two adjoining faces are parallel, then these

two triangular faces fall in the same plane, and together form a

trapezoid.

866. A Prismoid is a prismatoid whose bases have the same number of sides, and every corresponding pair parallel.

867. A frustum of a pyramid is a prismoid whose two bases are similar.

868. COROLLARY. Every three-sided prismoid is the frustum of a pyramid.

869. If both bases of a prismatoid become sects, it is a tetrahedron.

870. A Wedge is a prismatoid wh se lower base is a rectangle, and upper base a sect parallel to a basal edge.

H

871. The altitude of a prismatoid is any sect perpendicular to both bases.

872. A Cross-Section of a prismatoid is a section made by a plane perpendicular to the altitude.

RULE. Multiply one-fourth its altitude by the sum of one base and three times a cross-section at two-thirds the altitude from that base.

PROOF. Any prismatoid may be divided into tetrahedra, all of the same altitude as the prismatoid; some, as C FGO, having their apex in the upper base of the prismatoid, and for base a portion of its lower base; some, as O ABC, having base in the upper, and apex in the lower, base of the prismatoid; and the others, as A COG, having for a pair of opposite edges a sect in the plane of each base of the prismatoid, as AC and OG.

Therefore, if the formula holds good for tetrahedra in these three positions, it holds for the prismatoid, their sum.

I

In (1), call 7, the section at two-thirds the altitude from the base B1; then T, is a from the apex. Therefore, by 772,

T,

I

D1 = 2 (B1 + 3T;) = 2 (B, + 1B,) = 1aB,

:: D1

4

4

3

3

which, by 860, equals Y, the volume of the tetrahedron.

In (2), B, = 0, being a point, and T, is a from the apex;

:. T、 : B2 :: (}a)2 : a2, .. T1 = &B2,

a

a

I

:. D2 = 2 (B1 + 3 T1) = 2 (0 + ±B2) = ± aB2
Dz

4

4

In (3), let KLMN be the section T1.

3

3

= Y

▲ ANK : ▲ AGO :: AN2 : AG2 :: (a)2 : a2 :: I : 9.

▲ GNM : ▲ GAC :: GN2: GA2 :: (a)2 : a2

:: 4 : 9.

But the whole tetrahedron D, and the pyramid CANK may be considered as having their bases in the same plane, AGO, and the same altitude, a perpendicular from C;