SEGMENT
, in Geometry, is a part cut off the top of a figure by a line or plane; and the part remaining at the bottom, after the Segment is cut off, is called a frustum, or a zone. So, a
Segment of a Circle, is a part of the circle cut off by a chord, or a portion comprehended by an arch and its chord; and may be either greater or less than a semicircle. Thus, the portion ABCA is a Segment less than a semicircle; and ADCA a Segment greater.
The angle formed by lines drawn from the extremities of a chord to meet in any point of the arc, is called an angle in the Segment. So the angle ABC is an angle in the Segment ABCA; and the angle ADC, an angle in the Segment ADCA.
Also the angle B is said to be the angle upon the Segment ADC, and D the angle on the Segment ABC.
The angle which the chord AC makes with a tangent EF, is called the angle of a Segment; and it is equal to the angle in the alternate or supplemental Segment, or equal to the supplement of the angle in the same Segment. So the angle ACE is the angle of the Segment ABC, and is equal to the angle ADC, or to the supplement of the angle B; also the angle ACF is the angle of the Segment ADC, and is equal to the angle B, or to the supplement of the angle D.
The area of a Segment ABC, is evidently equal to the difference between the sector OABC of the same arc, and the triangle OAC on the same chord; the triangle being subtracted from the sector, to give the Segment, when less than a semicircle; but to be added when greater. See more rules for the Segment in my Mensuration, pa. 132 &c, 2d edition.
Similar Segments, are those that have their chords directly proportional to their radii or diameters, or that have similar arcs, or such as contain the same number of degrees.
Segment of a Sphere, is a part cut off by a plane.
The base of a Segment is always a circle. And the convex surfaces of different Segments, are to each other as their altitudes, or versed sines. And as the whole convex surface of the sphere is equal to 4 of its great circles, or 4 circles of the same diameter; so the surface of any Segment, is equal to 4 circles on a diameter equal to the chord of half the arc of the Segment. So that if d denote the diameter of the sphere, or the chord of half the circumference, and c the chord of half the arc of any other Segment, also a the altitude or versed sine of the same; then, 3.1416d2 is the surface of the whole sphere, and 3.1416c2, or 3.1416ad, the surface of the Segment.
For the solid content of a Segment, there are two rules usually given; viz, 1. To 3 times the square of the radius of its base, add the square of its height; multiply the sum by the height, and the product by .5236. Or, 2dly, From 3 times the diameter of the sphere, subtract twice the height of the frustum; multiply the remainder by the square of the height, and the product by .5236. That is, in symbols, the solid content is either ; where a is the altitude of the Segment, r the radius of its base, and d the diameter of the whole sphere.
Line of Segments, are two particular lines, so called, on Gunter's sector. They lie between the lines of sines and superficies, and are numbered with 5, 6, 7, 8, 9, 10. They represent the diameter of a circle, so divided into 100 parts, as that a right line drawn through those parts, and perpendicular to the diameter, shall cut the circle into two Segments, the greater of which shall have the same proportion to the whole circle, as the parts cut off have to 100.
