To find the area of a sector, find the arc (if only the angle be given), and multiply it by half the radius. To find the area of a segment contained between an aro and a chord, find the sector, and from it subtract the triangle formed by the terminal radii and the chord.
(Srfseef.) • To find the length of an arc, when the radius is not known, measure the chord of the arc, end the chord of its half; from eight times the chord of the half subtract the chord of the whole, and take one-third of the remainder. For an arc not exceeding 60% the error is less than the 7000th part of the whole.
The preceding are the principal rules of mensuration, of which the necessity occurs in the elementary parts of the subject : those which remain are connected with solid geometry, and the most essential are as follows : The number of cubic units in the content of a rectangular solid (or parallelopiped ; there is no shorter term for this most elementary form of solid figures, unless the reader will follow us in calling it a right solid) Is the product of the number of linear units in its three adjacent sides. The content of a prism or cylinder is the product of the number of square units in its base and the number of linear units in its altitude. The content of a pyramid or cone is one-third of the base multiplied by the altitude. The surfaces of a prism or pyramid must be found by computing those of their several faces. The surface of a common
cylinder (not including the bases) or of a right prism is the circum ference of the bow multiplied by the altitude; that of a common cone is half the product of the circumference of the base and the slant side.
The following formulas relate to the sphere. To find the aurfaco from the radius, multiply the square of the radius by 4 r, or the square of the diameter by w; to find the content from the radius, multiply the cube of the radium by 4 r : 3, or the cube of the diameter by w : 6 ; to find the radius from the surface, multiply the square root of the sur face by V (1 : 4 sr) ; to find the diameter, multiply the square root of the surface by (1 : w); to find the radius from the content, multiply the cube root of the content by .V(3 : 4w); to find the diameter, multiply the cube root of the content by .V(0 : sr); to find the surface from the content, multiply the cub* root of the square of six times the content by V to find the content from the surface, multiply the square root of the cube of the surface by 1 Va-. [SrlIEREJ For other formula, and methods, the detached articles on the different subjects may be consulted ; or Ilutton's or Bonnyeastle'e elementary works on mensuration. Many works on mensuration have been published of late years.