SEGMENT

(Lat.) In Geometry, a part cut off from a figure by a line or plane. The segment of a circle is a part of the area comprised between an arc and its chord; and segments of different circles are said to be similar when their arcs have the same ratio to the circumferences of their respective circles, or when they contain the same number of degrees. The area of a circular segment may be found u follows: Let P R be the chord of a circle whose centre is C, and let A B be the diameter perpendicular toP R, and meeting it in Q.; and putz = Aa,y=P Q,: = AP,r = A C the radius, and 7r = 3-14159, the ratio of the circumference to the diameter. Now the area of the segment P A R is evidently equal to the difference between the sector A P C R and the triangle P C R; but the area of the sector — n (the radius into half the arc), and the triangle = y () x); therefore the segment = r z — ry+ yz. When the radius of the circle and the chord of the segment (that is, r and y) are known, this expre ion is easily calculated from a table of sines, for the angle ACP is found from its siac = y-T-r; and when the angle is given the arc z is obtained from this proportion, z: ir r:: A C P: 180°. With respect to x, we have from the property of the circle x = ^/(rz— f). The superficial and solid contents of a segment of a sphere are found thus: Let S be the spherical surface of the segment of the sphere whose centre is C by the plane P Q. R. The symbols having the same values as above, the circumference of the circle P. O. R is 2 it y; and the differential of the surface is dS = 2irydz. But from the nature of the circle, ydz—rdx; hence d S = 2 7T rdx, and S = 2 nr i; so that the surface of the segment is found in terms of the radius of the sphere and the altitude of the segment. Let U denote the solid contents of the segment; then, since the area of the circle P Q. R is it y2, the differential of tr e segment is d U= tt y2 d x = 2 it r x d x — ir x2 d x; whence U = 7rz2(r — ix). [s. 1122]