SURFACE AND VOLUME OF SOLIDS.
Prism.— This includes any solid two of whose faces, known as the bases, are equal polygons situated in parallel planes and all the other faces, the lateral faces, are parallelograms. Thus in the following diagram (Fig. 7) the polygons ABC ... and A'B'C' ... are the bases and the parallelograms AA'B'B, BB'CC,... are the lateral faces The edges AA', BB', CC', ... are the The polygon UVW... lying in a plane perpendicular to one, and so to all, of the lateral edges a right section. The right section is equal, to the base if and only if the lateral edges are per pendicular to the base.
Far the lateral area a which is the sum of the lateral faces AA'B'B, BB'C'C ..., and the volume v of a prism we have the •formulas a= e(UV±VW-f-...), (XXX) v= es, (XXXI) v= hb, (XXXII) in which e= length of a lateral edge (AA' or BB' ...), s= area of a right section (UVW ...), h =height, i.e., distance between bases, b= area of a base.
The volume v of a truncated triangular prism, that is one with three lateral faces and whose bases are not parallel (therefore the lengths e,, e,, es of the lateral edges not equal) is v== is(ei+e2+ea). (XXXII!) Any truncated prism may be divided into truncated triangular prisms and its volume thus found. For a four-sided prism whose opposite lateral faces are parallel this gives v= I (e.-Fes)s, (XXXIV) where el ea are a pair of opposite edges.
Cylirider.— This includes any solid having two bases which are equal plane figures hounded by curved lines (such as circles, ellipses or irregular figures), and situated in parallel planes, and the rest of The surface of the solid such that it may be thought of as consisting of an infinite number of parallel straight lines. If we call the length of each of these (equal) parallel lines e, the formulas for the prism given above (XXX, XXXI, XXXII) apply also to the cylinder, namely, the lateral area a and the volume v are a =.1= ep, (XXXV) es, (XXXVI) hb, (XXXVII) where p is the length of the boundary, and s is the area of the section of the cylinder by a plane perpendicular to one of the lines lying on the lateral surface.
If the right section of a cylinder be a circle, the base, unless a circle, is an ellipse and its area is found by formula (XXVI). If in such
a circular cylinder the bases are not parallel and the distance between the centres of the bases is I, then the area a and the volume v are 1p, (XXXVIII) 27rsl, (XXXIX) v= Is, (XL) =awl, r =3.1416, (XLI) where r is the radius of the right section. These formulas are equivalent to (XXXV) and (XXXVI) when the bases are parallel, since then e.
Pyramid and Cone.—The volume v of any solid, such as those in Fig. 8, whose sur face consists of a base, which is a plane figure of any kind, triangle, polygon, circle, ellipse, etc., and of triangles or curved areas which may be thought of as composed wholly of straight lines, joining the boundary of the base to some point (the vertex) not in the same plane as the base, is v=ribh, (XLII) where b is the area of the base and h is the perpendicular distance (height) of the vertex from the plane of the base.
For a right circular cone. that is one in which the base is a circle and the line joining the vertex to the centre of the base is perpen dicular to the plane of the base, the area a of the curved (lateral) surface is a=trre, (XLIII) where r is the radius of the base and e is the distance from the vertex to any point in the circumference of the base.
Prismatoid.— The volume v of any solid, whose total surface consists of two plane figures (the bases), of any character whatever, lying in parallel planes, and a lateral surface made up of triangles, trapezoids, or curved portions which may be thought as made up of straight lines joining the boundaries of the bases, is (b1-1-bt-I-4nt), (XLIV) where h is perpendicular distance (height) be tween the bases, b, and b, are the areas of the bases, and m is the area of a section of the solid by a plane parallel to and half-way between the planes of the bases; or v=ih(b 3q), (XLV) where b is one of the bases and q is the area of a section parallel to and two-thirds of the way from, that base to the other.
Solids of Revolution. Sphere.— The area s of the surface and the volume v of a sphere of radius r are s=47rrs, 47r= 12.56637 . . . (XLVI) 47r v =—rs,