MENSURATION 17 (7) Radius of Circumscribed Circle. Let AD (fig. 12)=j? the perpendicular from A on the side BC, and AE = 2R the diameter Fig. 11. of the circle, then (Eucl. vi. C) we have Fig. 12. therefore 2R x ap = abc . Now c<p = 2A, where A denotes the area of ABC ; hence R = ^- c alc Example. Let <x = 13, J = 14, andc = 15; then r will be found to be 4, r a 10, r b 12, r c 14, and R, 8*. SECTION II. PLANE FIGURES CONTAINED BY CURVED LINES. A. The Circle. 22. Circumference of a Circle. If we inscribe in any circle a regular polygon of n sides, and also circumscribe a regular polygon of the same number of sides, it is clear that the perimeter of the circle is intermediate between the perimeters of the inscribed and circumscribed polygons, and that the difference between the peri meters of the inscribed and circumscribed polygons can be made as small as we please by sufficiently increasing n. A similar state ment holds with reference to the areas of the circle and the in scribed and circumscribed polygons. With the above assumptions it is easily proved that the circumference of a circle bears a constant ratio to its diameter. Hence we have Circumference = C = constant x radius = constant x r. It is usual to denote this constant by lir, and therefore C = 2irr=ird, where d is the diameter of the circle. 23. Numerical Value of TT. The constant TT being, as can be easily proved, an interminable decimal, we can only approximate to its value, but this we can do to any degree of accuracy we please. If s and or denote respectively a side of the inscribed and circum scribed polygons of n sides, and s and cr a side of the inscribed and circumscribed polygons of 2n sides, it can easily be shown that / i i rs (V) a- = /-= = > Vr>-(is ) 2 where r is the radius of the circle. If we take r= we find, by means of these formulte, and by assuming the value of s when n = 6, that the perimeter of inscribed polygon of 96 sides = 3 140 .... , and the perimeter of circum scribed polygon of 96 sides =3 142. . . . From this we learn that the circumference of the circle, in this case TT, is greater than 3140 .... and less than 3 142 . . . . , and therefore as far as the second place of decimals w-8 14. By taking greater and greater values of n we obtain closer and closer approximations to IT. The preceding method for approximating to the value of IT is the simplest afforded by elementary geometry, and is also the oldest ; but better and more rapid methods are furnished by the higher mathematics. The calculation of TT has been carried to 707 places of decimals, the following being the first 20 figures in the result: 314159265358979323846. For all practical purposes it is sufficient to take 24. The following table contains the functions of TT that are of most frequent occurrence in mensuration : Number. Logarithm. Number. Logarithm. T 3-1415927 0-4971499 T 2 9-8696044 0-9912997 2;r 6-2831853 0-7981799 - 47T 12-5663706 1-0992099
0-0168869 2-2275490 JT 1-5707963 0-1961199 BTT" iT 1-0471976 0-0200286 -V/T 1-7724539 0-2485750 JT 0-7853982 1-8950899 0-5235988 1-7189986 3/
T
1-4645919 0-1057166 IT 0-3926991 T -5940599 AT 0-2617994 1-4179686 1 0-5641896 1-7514251 4-1887902 0-6220886 /TT IT 180 1 0-0174533 2-2418774 2 vv 1-1283792 0-0524551 7T 0-3183099 T-5028501 27/T 0-2820948 1-4503951 4 - 1-2732395 0-1049101
- J
1-2407010 0-0936671 1 0-0795775 5-9007901 3/1 V4JT 0-C203505 I-792G371 180 57-2957795 1-758122G logen- 1-1447299 0-0587030 25. Units of Angular Measurement. In measuring lines we select some line of constant length as the standard or unit ; simi larly in measuring angles we require to take some angle of constant magnitude as unit angle. The right angle is by its nature the simplest unit angle, but, for convenience, we take the isVth. part of a right angle for unit, and call it a degree, which is subdivided into sixty equal parts called minutes, and these again into sixty equal parts called seconds. For theoretical purposes we define the unit angle to be the angle subtended at the centre of a circle by an arc equal to the radius. This angle we call a "radian." In many treatises the radian measure of an angle is called the circular measure. 26. The radian is a constant angle. Let OA (fi ( = r, then AOB = radian, and let AOD = 90; then arc AD = J x 2irr = irr ; and, since angles at the centre of a circle are proportional to the arcs on which they stand (Eucl. vi. 33), number of degrees in radian AOB_ AB_ r _ 2 number of degrees in AOD AD -nr -n therefore number of degrees in radian = 90 x - 57 2957795 = constant. 7T 27. Number of Radians in any Angle. Let AOC (fig. 13) bo any angle, AOB the radian, and AC = s ; then number of radians in AOC _ AC s_ . one radian AB r therefore number of radians in AOC = . r If AOC = 90, then s = ^irr, and number of radians = ir ; there are thus TT radians in two and 2ir in four right angles. When r = l we have number of radians = s, and hence in some treatises for the number of radians in an angle we find the length of the arc given. 28. To transfer from degrees to radians and conversely. Let x denote the number of degrees in an angle, and the number of radians in the same, then, since x = 180 TT 29. The following table contains the values of for values of x up to 180, and also for minutes and seconds. p a Q Radian. | Radian. 2 tc & Radian. Minutes. Radian.
Seconds. Radi an. 000 1 0174533 61 1 -064(5508 121 2-111S4S4 1 002909 1 0048 2 0349066 62 1-0821041 122 2-1293017 2 005818 2 0097 3 0523599 63 1-0995574 123 2-1467550 3 008727 3 0145 4 0698132 64 1-1170107 124 2-1G42083 4 011636 4 0194 5 0872665 65 1-1344640 125 2-1S16616 5 014544 5 0242 6 1047198 66 1-1519173 126 2-1991149 6 017453 6 0291 7 1221730 67 1-1693706 127 2-2165682 7 020362 7 0339 8 1396263 68 1-1868239 l-. s 2-2340214 8 023271 8 0388 9 1570796 69 1-2042772 129 2-2514747 9 0-26180 9 0436 10 1745329 7D 1-2217305 l:)<> 2-2689280 10 029089 10 0485 20 3490659 80 1-3962C34 140 2-4434610 20 058178 20 0970 30 5235988 90 1-5707963 150 2-6179939 30 087266 30 1454 40 6981317 100 1-7453293 160 2-7925268 40 11R355 40 1939 50 8726646 110 1-9198622 170 2-9670597 50 145444 i 50 2424 60 1-0471976 120 2-0943951 180 3-1415927 60 174533 60 2909
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