MENSURATION OF GEOMETRICAL FIGURES I. Areas of Plane Rectilinear Figures.—(i.) We begin with some simple plane figures. The square, rectangle, parallelogram and triangle may all be regarded as particular cases of the trapezium (U.S.A. usage, trapezoid, the word "trapezium" being generally dropped), which is a quadrilateral with two parallel sides. If the sides are a and b, and the distance between them is h, the area of the trapezium is 11-1(a+b). In the case of the triangle, for instance, b is zero, so that the area is lha, i.e., half the product of a side by the perpendicular distance of the oppo site angle from it.
(ii.) If the two parallel sides of a trapezium are at right angles to one of the remaining sides, the figure is called a right trapezium. The two parallel sides are the sides, the side at right angles to them is the base, and the fourth side is the top. The area is the product of the base by the mean of the two sides.
(iii.) A figure such as ABCDEFSM (fig. I), bounded by a base MS, two sides MA and SF at right angles to the base, and a rectilinear top ABCDEF, is called a trapezoid. (In U.S.A., generally referred to as a general quadrilateral, "trapezoid" being used for "trapezium.") It can be split up into a series of right tra pezia by drawing perpendiculars BN , CP, DQ, ER. One or both of the extreme trapezia may be a triangle. This would be the case, e.g., if in fig. i F and S coincide.
(iv.) The area of any rectilinear figure may be found in various ways, as for example: (a) The figure may be divided into triangles. The quadri lateral, for instance, can be treated as made up of two tri angles, and its area is the prod uct of half the length of one diagonal by the sum of the per pendiculars drawn to this diag onal from the other two angular points.
(b) The figure may, by draw ing a straight line through it, be divided into two trapezoids. 2. Surfaces and Volumes of Solids.—The solid figures with which we are concerned may be grouped as of various types, some coming under more than one type.
(i.) The prism is a figure bounded by two parallel planes, all sections by planes parallel to these planes being congruent figures with corresponding points lying on parallel lines. The
length of any one of these lines is the length of the prism. The section at right angles to the lines is the cross-section. This definition includes the cube, the parallelepiped and the cylinder. The surface of a prism (excluding the two ends) is = (perimeter of cross-section) X length; the volume is = (area of cross section) X length.
(ii.) The prismoid (or prismatoid) may be defined as a solid figure with two parallel plane rectilinear ends, each of the other (i.e., the lateral) faces being a triangle with an angular point in one end of the figure and its opposite side in the other. Two ad joining faces in the same plane may together make a trapezium. More briefly, the figure may be defined as a polyhedron with two parallel faces containing all the vertices. The definition covers prism, pyramid and frustum of a pyramid; the cylinder, cone and frustum of a cone may be regarded as limiting cases.
The volume of a prismoid is 4--h(A +4C+B), where la is the perpendicular distance between the two ends, A and B are the areas of the ends, and C is the area of a section parallel to the ends and midway between them. This is the prismoidal formula: it applies to other figures than the prismoid, e.g., to the sphere (see §13).
In the case of a cylinder, A = B=C; in the case of a pyramid or a cone A is o, and C is one-fourth of B, so that the volume is 31B; in the case of a frustum of a pyramid or a cone CI is the mean of A 'and Bi, and the volume is thereforelli[A -I- (A B)i Bj.
(iii.) The surface (excluding the ends) of a prismoid is develop able, i.e., it can be unrolled flat on a plane. Hence the area of the curved surface of a right circular cone is = 1(slant height) X (perimeter of base). • (iv.) The sphere is not a prismoid, but the prismoidal formula is applicable to its volume. If a is the radius of a sphere, then Volume of sphere = Ire= 3 vol. of circumscribing cylinder, Sufface of sphere = = curved surface of circumscribing cylinder =3- total surface of circumscribing cylinder.