(13) The chord of an arc is 24 in., and the height of the arc is 9 in.; find the diameter of the circle. (14) The chord of an arc is 4 ft. 4 in., and the height of the arc is 1 ft. 1 in.; find the diameter of the circle. (15) The span (chord) of a bridge, the form of which is an arc of a circle, is 78 ft., and its height above the stone piers is 12 ft.; find with what radius it was described. (16**) The span of a bridge, the form of which is an arc of a circle, being 96 ft., and its height being 12 ft., with what radius was it described? (17) The height of the arch of a bridge, the form of which is an arc of a circle, is 24 ft., and the radius with which it is described is 312 ft.; find the span of the arch. XIV. THE ARC OF A CIRCLE. Definitions.—An arc of a circle is any part of the circum The arc ACB bears the same ratio to the circumference of the circle that the number of degrees in the angle AEB does to 360°. RULES. (1) To find the length of the arc of a circle, when the number of degrees in the angle subtended by the arc at the centre, and the radius of the circle, are given. The radius of the circle being given, find its circumference by Rule 1, Prob. XI. Then we have the following proportion for finding the length of the arc :— 360° number of degrees subtended by arc at centre circumference required length of arc. (2) To find the number of degrees in the angle subtended by the arc at the centre of a circle, when the radius of the circle and the length of the arc are given. The radius of circle being given, find its circumference by Rule 1, Prob. XI. Then we have the following proportion for finding the number of degrees subtended by arc at the centre:- Circumference of circle: length of arc:: 360° : number of degrees subtended by arc at centre. (3) To find approximately the length of an arc of a circle less than a semicircle, when the chord of the arc and the chord of half the arc are given. From 8 times the chord of half the arc subtract the chord of the whole arc; divide the remainder by 3, and the quotient is the length of the arc, nearly. Note 1.-The answers obtained by Rule 3 will not be quite correct, but they will be sufficiently correct in all cases when the arc is less than a semicircle. Note 2.-When it is required to find the length of an arc AFB, greater than a semicircle, it will be better to find, first, the length of the remaining arc ACB by the preceding rules. Then the length of the arc AFB circumference of circle-arc ACB. Note 3.-If it is desirable to find the length of the arc with still greater accuracy than is possible by Rule 3, we may proceed thus: If the diameter is not given, find it by the rules given in the last chapter. Then subtract of the height of the arc from the diameter; divide of the height by the remainder; and to the quotient add 1. Then multiply this sum by the chord of the arc, and the product is the length of the arc, very nearly. Example 1.-The circumference of a circle is 50 inches, and the angle subtended at the centre by the arc is 30°; find the length of the arc. By Rule 1, we have: 360° 30°:: 50 in. : required length of the arc; and from this proportion we get 4 in.-length of the arc. Example 2.-The radius of a circle is 28 inches, and the angle subtended at the centre by the arc is 45°; find the length of the arc. First, the circumference of the circle, by Rule 1, Prob. XI., is 56 x 22=176 in. Then we have, by Rule 1: 360° 45°:: 176 inches required length of the arc. From this proportion we get 22 in.-length of the arc. Example 3.-The arc of a circle is 10 in.; the radius of the circle is 12 in. Find the angle subtended at the centre by the arc. The circumference of the circle=24 x 29=75 in. Then by Rule 2, we have: 75 inches: 10 inches :: 360° : required angle subtended by the arc. From this proportion we obtain 478, the angle subtended at the centre. Example 4.-The chord of an arc less than a semicircle is 20 ft., and the chord of half its arc is 10.198 ft.; find the length of the arc. By Rule 3: 10.198 × 8=81 584 Subtract 20 3)61.584 20-528 ft., length of arc, nearly. Example 5.-Find the length of the arc of a circle less than a semicircle; the chord of the arc is 20 ft., and the diameter of the circle is 29 ft. In the right-angled triangle ADE (see the figure) we have DE=√AE2-AD2=√14·52-102=10·5 ft.; and CD CE-DE-14.5-10.5=4 ft. Again, in the right-angled triangle ADC, we have AC, chord of half the arc, 116 10.77 + ft. = Then, by Rule 3, we have: 10.77 x 8=86.16 Formulæ Subtract 20 AD2+DC2102 +42= 22.05, length of the arc, nearly. I. 360° number of degrees subtended by arc at centre:: circumference of circle required length of arc. II. Circumference of circle: length of the arc 360°: number of degrees subtended by arc at centre. 8 AC-AB III. Arc.= E 3 EXAMPLES. (1) The radius of a circle is 10.5 in.; the angle subtended by the arc at the centre is 60°; find the length of the arc. (2) The diameter of a circle is 35 in.; the angle subtended by an arc at the centre is 36°; find the length of the arc. (3) The arc of a circle is 6 ft. 5 in.; the radius of the circle is 8 ft. 2 in.; find the angle subtended at the centre by the arc. (4) The radius of a circle is 7 in.; the length of an arc is the same; find the angle at the centre subtended by the arc. (5) The radius of a circle is 5 ft. 3 in.; find the whole perimeter of a sector, the angle of which is 45°. (6) The arc of a circle is 5 ft. 6 in.; the angle subtended by the arc at the centre is 72°; find the radius of the circle. (7) The arc of a circle is 26.4 ft.; the angle subtended by the arc at the centre is 36°; find the radius of the circle, of which the arc is a part. (8) The radius of a circle is 84 ft.; the angle subtended at the centre by an arc is 11° 15'; find the length of the arc. (9) The chord of an arc is 32 in.; the radius of the circle is 34 in.; find the length of the arc. (10) The chord of an arc is 2 ft. ; the radius of the circle is 1 ft. 3 in.; find the length of the arc. (11) The chord of an arc is 64 ft. ; the height of the arc is 24 ft.; find the length of the arc. (12) The chord of an arc greater than a semicircle is 70 ft.; the radius of the circle is 37 ft. ; find the length of the arc. (13) The diameter of a circle is 106 ft.; find the lengths of the two arcs into which a chord of 90 ft. would divide it. (14) The chord of an arc is 240 ft.; the height of the |