SEGMENT (part cut off), a term which, In its general sense, needs no explanation. It is, in mensuration. most frequently applied to the part cut off from a circle by a chord, and the measurement of this segment of a circle is the only point for which reference is likely to be made to the word Let • a be the segment of a circle, and o and D the middle points of its arc and chord. The segment AO a Is easily expressed by the angle which the arc subtends at the centre : if this angle (measured in the theoretical units [ANGLE]) be e, and the radius r, the number of square unite in the segment is rs (0—sin. e).
But when a segment is actually to be measured in practice, it usually happens that the radius is not given, and the circle is too large to measure it conveniently. In that case the middle point o must be found, and A a and a C must bo measured, as also c D. This being done, the length of the are A a can be found with great exactness from the formula " one third of the excess of eight times e c over a a, or (8 a C Br This formula errs only about one foot out of 80 (always giving the arc a little too small) in a whole semicircle, and the error diminiehea nearly as the fourth power of the arc ; thus at half a semicircle the error is about one foot out of 80, or 1280 feet ; at ono-third of a semicircle it is only about one foot out of 80, or 64S0 feet, and so on. Another formula of the same sort, but so close
to the truth that no measurements could ever be taken in practice sufficiently exact to make its inaccuracy appreciable, is the following find E the middle point of a o, and measure A then the arc is very nearly (but a little less than) ea+ 256 A E-40 A Ca 45 The error here is less than one foot out of 390 on the whole circle, and diminishes with the sixth the arc.
Taking the arc from one or other of these methods, the area of the segment is then to be found as follows :—Determine R, the radius, from A $4-9 C D, and compute R X Aro—A B (R.-0 2 which gives the area required. This formula may be reduced to o2(2 D Thus (to take an instance of I3onuycastle's) if a nas.12, A C 13, whence c na.5 (all feet), we have for the area of the segment x 169 x square feet. 5 Another approximate rule is (giving somewhat too little)