Elements of geometry, theoretical and practical; including constructions of the right line and by the circle together with the mensuration of all elementary plane figures and solids - Softcover
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1836 Excerpt: ...then r=10 and the area=314.15926 PROBLEM XII. 84. To find the area of a sector of a circle, having the arc and radius given. The area=ħar 12, in which a denotes the arc of the sector. Example. Let a be one tenth of the circumference, corresponding to r=6 feet. The circumference c=37.6992, the arc a=3.76992; and the area=3x3.76992=11.30976 square feet. PROBLEM XIII. 85. To find the arc of a circular segment of which the chord and height are given. TIL Cm The arc a=----, nearly 13, in which h is the height of the segment, c the chord, and m--3r--iA, or 3 r--J-A, according as h is greater or less than half the radius r. For let A B A' be a segment, of which the semiarc A B is one fifth of the circumference described from O with the radius O B = 1. Then O E=.309017 being the semichord of a regular inscribed decagon, there will result AE or£c=.951057, BE or h=.690983, and B C or m=2.827255. By virtue of these values, the first formula gives a=2.51738, which is nearly equal to the true value a=2.51328. The same formula in the extreme case of A=l gives a=2T2, one of the approximate values of 7T; and these cases sufficiently show its application and accuracy. As to the second formula, let A A' be the side of a regular inscribed decagon. Then c=.618034, A=.048943, m=2.990211, and a=.628318, which is extremely near the true value of the arc. Remark. From either of the. formula here given, there results a=c(l+iA), when h is considered sufficiently small to neglect its square, and higher powers. According to this supposition, h= c3, and therefore a=c-f 5T c3, which may be employed to determine the arc of a segment of which the height is small relatively to its chord. PROBLEM XIV. 86. To find the area of the segment of a circle, of which the radius, chord, and...
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