Method 2. Multiply the diameter by 31416, and the product will be the circumference. Or divide the circumference by 3·1416, and the quotient will be the diameter. This inethod comes nearer to the truth than the foregoing, but for practical purposes, Method 1 will be sufficiently near. Prob. 10. The chord and height of a segment being given, to find the chord of half the arc. To the square of the half chord, add the square of the versed sine, and the square root of the sum will be the length of the chord of half the arc. L. The chord AC being 48 feet, and the versed sine DB 18 feet, what is the length of the chord AB or BC of half the are? 00 Prob. 11. To find the length of any arc of a circle, the half chord and chord of the whole arc being given. : Subtract the chord of the whole are from double the chord of the half arc add one-third of the remainder to the double chord of the half arc, and the sum will be nearly equal to the length of the arc. Ex. 1. What is the length of the arc ABC, of which the chord AC is 48, and the half chord AB or BC is 30? 2 x 30-60 the double chord of the half ar -48 the chord of the whole arc. - 3)12 D 64 the length of the arc required. Prob. 12. The chord and height of a segment being given, to find the radius of the circle. To the square of the half chord, add the square of the versed sine ; divide the sum by twice the versed sine, and the quotient will be the radius of the circle, when it is less than a semicircle. Ex. The chord AC of a segment ABC being 48 feet, and the versed sine 18 feet, what is the radius of the circle? See the diagram to Example Problem 10. Prob. 13. Given any two paralle. chords in a circle, and their as tance, to find the distance of the greater chord from the centre. To the square of the distance between the chords, add the square of half the lesser chord. The difference between this sum, and the square of half the greater chord divided by twice the distance of the chords, will give the distance of the centre from the greatest chord. Er. Suppose the greater chord CD is 48 feet, and the lesser AB 30, their distance FG 13 feet, how far is the centre E from the greater chord CD? Prob. 14. Given a chord of a circle and its distance from the cen tre, to find the radius of the circle. To the square of the half chord, add the square of the distance from the centre, and the square root of the sum will be the radius required Ex. Given the chord CD 48 feet, and its distance EF from the centre 7 feet, required the radius of the circle. See the figure to the Example Problem 13. 24 and 24x24=576 625 (25 the radius 45)225 225 Prob. 15. Given any two parallel chords in a circle, and the distance between them, to find the perpendicular height from the middle of either chord to the circumference. Find the nearest distance of the greater chord from the centre, by Problem 13, and find the radius of the circle by Problem 14, add the distance between the two parallel chords, and the distance between the greater chord, and the centre of the circle together: this sum being taken from the radius, will give the perpendicular height from the middle of the lesser chord, to the circumference or height of the lesser segment; to the lesser segment, add the distance between the parallel chords, and the sum will be the height of the greater seg ment. Ex. Given the greater chord CD 48 feet, and the lesser chord AB 30 feet, their distance EG 13 feet, required the distance GH perpendicular from the middle of AB to the circumference. See the figure to the Example Problem 13. The distance from the centre to the greater chord, will be found to be 7 feet, by Problem 13, and the radius 25 feet, by Problem 14. 13+7=20 and 25-20=5 feet, height of the lesser segment. Then 13+5=18 the height of the greater segment. Prob. 16. To find the area of a circle, the diameter being given. Method 1. Multiply half the circumference by half the diameter, and the product will be the area. Ex. 1. What is the area of a circle whose diameter is 28 feet, and its circumference 88 feet? 44 Y=14 176 616 feet, the area required. Method 2. Multiply the square of the diameter by *7854, and the product will be the area. Ex. What is the area of a circle whose diameter is 3f. 6i. ? 3.5 175 105 12 25 square of the diameter. *7854 4900 6125 9800 8575 9.621150 the answer. In common practice, multiply the square of the diameter when given in feet, inches, &c. by 9i. 5ii. Ex. What is the area of a circle whose diameter is 3f. i.? Method 3. When the circumference is given. Multiply the square of the circumference by '07958, and the product will be the area. Ex. What is the area of a circle when the circumference is 88 feet? 88 88 704 704 7744 square of the circumference. '07958 61952 38720 69696 54208 616 26752 the area of the circle. Prob. 17. To find the area of a sector of a circle. Multiply the radius, or half the diameter, by half the length of the arc of the sector, and the product will be the area, Fr. 1. What is the area of a sector ABC, the arc BC being 3f. 6i. and the radius AB or AC 6f. 21. ? Prob. To find the area of the segment of a circle, the chord and height of the arc being given. Find the length of the arc ABC by Prob. 11, and the radius of the circle by Prob. 12, the area of the sector ABCE by Prob. 17. Subtract the area of the triangle AEC, fend by Prob. 2, from the area of the sector, and the remainder will be the area of the segment. |