1. Required the area of the regular pentagon ABCDE whose side AB, or BC, &c. is 25 feet, and perpendicular OP 17.2 feet. D A P B 25 x 5 Here =62.5=half perimeter ; and 62.5 x 17.2= 2 1075 square feet=area required. 2. Required the area of a hexagon whose side is 14.6 feet, and perpendicular 12.64. - Ans. 553.632 square feet. 3. Required the area of a heptagon whose side is 19.38, and perpendicular from the centre 20. Ans. 1356.6. 4. Required the area of an octagon whose side is 9.941 and perpendicular 12. Ans. 477.168 a PROBLEM VIII. To find the area of a regular polygon, when the side only is given. RULE.* Multiply the square of the side of the polygon by the nuinber standing opposite to its name in the following table, and the product will be the area. * Demon. The multipliers in the table are the areas of the poly. gons to which they belong when the side is unity, or 1. Whence as all regular polygons, of the same number of sides, are No. of Names. Multipliers. 3 Trigon or equil. A Nonagon Duodecagon 0.433013_ 9.365640- 1 . tang similar to each other, and as similar figures are as the squares of their like sides, (Euc. VI. 20.) 12: multiplier in the table :: square of the side of any polygon: area of that polygon; or which is the sarne thing, the square of the side of any polygon X by its tabular number israrea of the polygen. Q. E. D. The table is formed by trigonometry, thus: As radius BP X tan. LOBP LOBP :: BP (1): Po= 1 tang. LOBP: radius whence OP X-BP=1 tang. LOBP = area of the A AOB; and 4 tang. LOBP X number of sides=tabular number, or the area of the polygon. The angle obp, together with its tangent, for any polygon of not more than 12 sides, is shown in the following table. No. of Angle Tangents. ОВР. 54° ? Trigon 30° .57735+= 3 1.37638+=13V5| Heptagon 64°, 2.07652+ 8 Octagon 67° 2.41421+=1+12 6401 2.74747+ 72° 13.07768+=15+275 EXAMPLES. 1. Required the area of a pentagon whose side is 15. The number opposite pentagon in the table is 1.720477. Hence 1.720477x152=1.720477 X 225=387.107325= area required. 2. The side of a hexagon is 5 feet 4 inches ; what is the area? Ans. 73.9. 3. Required the area of an octagon whose side is 16. Ans. 1236.0773. 4. Required the area of a decagon whose side is 20.5. Ans. 3233.4913. 5. Required the area of a nonagon whose side is 36. Ans. 8011.6439. 6. Required the area of a duodecagon whose side is 125. Ans. 174939.875, PROBLEM IX. The diameter of a circle being given, to find the circumference; or, the circumference being given, to find the diameter. RULE.* Multiply the diameter by 3.1416, and the product will be the circumference, or * The proportion of the diameter of a circle to its circumference has never yet been exactly ascertained. Nor can a square or any other right lined figure, be found, that shall be equal to a given circle This is the celebrated problem called the squaring of the circle, which has exercised the abilities of the greatest mathematicians for ages, Divide the circumference by 3.1416, and the quotient wil. oe the diameter. Note 1.-As 7 is to 22, so is the diameter to the circumference; or as 22 is to 7, so is the circumference to the di ameter. 2. As 113 is to 355, so is the diameter to the circum ference; or, as 352 is to 115, so is the circumference to the diameter. and been the occasion of so many disputes. Several persons of con. siderable eminence, have, at different times, pretended that they hag discovered the exact quadrature; but their errors have soon been de. tected, and it is now generally looked upon as a thing impossible to be done. But though the relation between the diameter and circumference cannot be accurately expressed in known numbers, it may yet be ap. proximated to any assigned degree of exactness. And in this manner was the problem solved by the great Archimedes, about two thousand years ago, who discovered the proportion to be nearly as 7 to 22, which is the same as our first note. This he effected by showing that the perimeter of a circumscribed regular polygon of 192 sides, is to the diameter in a less ratio than that of 377 to 1 and that the perime. ter of an inscribed polygon of 96 sides is to the diameter in a greater ratio than that of 3 | to 1, and from thence inferred the ratio above mentioned, as may be seen in his book De Dimensione Circuli. The same proportion was also discovered by Philo Gedarensis and Apollo. nius Pergeus at a still earlier period, as we are informed by Eutocius in his observations on a work called Ocyteboos. The proportion of Vieta and Metius is that of 113 to 355, which is something more exact than the former, and is the same as the second note. This is a very commodious proportion : for being reduced into deci. mals, it agrees with the truth as far as the sixth figure inclusively. It was derived from the pretended quadrature of a M. Van Eick, which first gave rise to the discovery. But the first who ascertained this ratio to any great degree of ex. actness was Van Ceulen, a Dutchman, in his book, De Circulo et Adfcriptis. He found that if the diameter of a circle was 1, the circum. ference would be 3.141592653589793238462643383279502884 nearıy. wnich is exactly true to 36 places of decimals, and was effected by the continual bisection of an arc of a circle, a method so extremely EXAMPLES. a 1. If the diameter of a circle be 17, what is the circum rence ? Here 3.1416 x 17=53.4072=circumference. a 2. If the circumference of a circle be 354, what is the diameter ? 354.000 Here =112.681=diameter. 3.1416 3. What is the circumference of a circle whose diameter is 40 feet? Ans. 125.6640 4. What is the circumference of a circle whose diame. ter is 12 feet? Ans. 37.6992. 5. If the circumference of the earth be 25000 miles, what is its diameter? Ans. 7958 nearly. 6. The base of a cone is a circle ; what is its diameter when the circumference is 54 feet? Ans. 20.3718. a troublesome and laborious that it must have cost him incredible pains. It is said to have been thought so curious a performance, that the numbers were cut on his tomb-stone in St. Peter's Church-yard, at Leyden. This last number has since been confirmed and extended to double the number of places, by the late ingenious Mr. Abraham Sharp, of Little Horton, near Bedford, in Yorkshire. But since the invention of Fluxions, and the Summation of Infinite Series, there have been several methods discovered for doing the same thing with much more ease and expedition. The late Mr. John Machin, Professor of Astronomy in Gresham College, has by these means given a quadrature of the circle which is true to 100 places of decimals; and M. de Lagny, M. Euler, &c. have carried it still further. All of which proportions are so extremely near the truth, that, except the ratio could be completely obtained, we need not wisb for a greater degree of accuracy. |