To compute the radius of a circumscribing circle when the length of a side only is given. RULE. Multiply the length of a side of the polygon by the number in column B. EXAMPLE. What is the radius of a circle that will contain a hexagon, the length of one side being 5 inches? Ans. 5 × 1 = 5 inches. To compute the length of a side of a polygon that is contained in a given circle, when the radius of the circle is given. RULE. — Multiply the radius of the circle by the number opposite the name of the polygon in column C. EXAMPLE. - What is the length of the side of a pentagon contained in a circle 8 feet in diameter ? Ans. 8 ft. diameter ÷ 2 = 4 ft. radius, 4 × 1.1756 = 4.7024 ft. To compute the radius of a circle that can be inscribed in a given polygon, when the length of a side is given. RULE. Multiply the length of a side of the polygon by the number opposite the name of the polygon in column D. EXAMPLE. - - What is the radius of the circle that can be inscribed in an octagon, the length of one side being 6 inches. Ans. 6 x 1.2071 = 7.2426 inches. Circles. To compute the circumference of a circle. - RULE. Multiply the diameter by 3.1416; or, for most purposes, by 34 is sufficiently accurate. EXAMPLE. diameter ? - What is the circumference of a circle 7 inches in Ans. 7 x 3.1416 = 21.9912 inches, or 7 × 3 = 22 inches, the error in this last being 0.0088 of an inch. To find the diameter of a circle when the circumference is given. RULE. Divide the circumference by 3.1416, or for a very near approximate result multiply by 7 and divide by 22. To find the radius of an arc, when the chord and rise or versed sine are given. RULE.-Square one-half the chord, also square the rise; divide their sum by twice the rise; the result will be the radius. EXAMPLE. The length of the chord ac, Fig. 30, is 48 inches, and the rise, bo, is 6 inches. What is the radius of the arc ? To find the rise or versed sine of a circular arc, when the chord and radius are given. RULE. - Square the radius; also square one-half the chord; sub. tract the latter from the former, and take the square root of the remainder. Subtract the result from the radius, and the remainder will be the rise. -A given chord has a radius of 51 inches, and a chord of 48 inches. - To compute the area of a circle. RULE. = 5145 6 inches = rise, - Multiply the square of the diameter by 0.7854, or mul tiply the square of the radius by 3.1416. EXAMPLE. - What is the area of a circle 10 inches in diameter ? Ans. 10 x 10 × 0.7854 = 78.54 square inches, or 5 × 5 × 3.1416 = 78.54 square inchies. The following tables will be found very convenient for finding the circumference and area of circles. AREAS AND CIRCUMFERENCES OF CIRCLES. (Advancing by Tenths.) Diam. Area. Circum. Diam. Area. Circum. Diam Area. Circum. 226.9801 .2 232.3522 .3 53.4071 22.0 380.1327 69.1150 27.0 572.5553 .5 397.6078 70.6858 .5 593.9574 86.3938 68799 .6 598.2849 86.7080 .7 602.6282 68789 84.8230 1234 576.8043 85.1372 581.0690 85.4513 585.3494 85.7655 415.4756 72.2566 28.0 .1 419.0963 72.5708 .1 620.1582 88.2788 254.4690 56.5486 23.0 .2 72.8849 .3 426.3848 73.1991 .4 430.0526 73.5133 .5 433.7361 73.8274 624.5800 88.5929 629.0175 88.9071 .4 633.4707 89.2212 .3 .5 637.9397 89.5354 .6 437.4354 74.1416 .6 642.4243 58788 .8 277.5911 59.0619 .8 444.8809 74.7699 .9 280.5521 59.3761 .9 448.6273 75.0841 19.0 .1 .2 289.5292 60.3186 68799 .8 651.4407 90.4779 .9 655.9724 90.7920 283.5287 59.6903 24.0 452.3893 75.3982 29.0 292.5530 60.6327 .4 295.5925 60.9469 298.6477 61.2611 .6 301.7186 61.5752 .7 304.8052 61.8894 .8 307.9075 62.2035 .9 311.0255 62.5177 1234 .3 660.5199 91.1062 .1 665.0830 91.4203 .2 669.6619 91.7345 674.2565 92.0487 .4 678.8668 92.3628 1234 44588 .5 683.4928 92.6770 .6 688.1345 92.9911 .7 692.7919 93.3053 .8 697.4650 93.6195 .9 486.9547 78.2257 .9 702.1538 93.9336 |