By this table may be calculated the area of any other regular polygon, of the same number of sides with one of these. For the areas of similar polygons are as the squares of their homologous sides. (Euc. 20. 6.)* To find, then, the area of a regular polygon, multiply the square of one of its sides by the area of a similar polygon of which the side is a unit. Ex. 1. What is the area of a regular decagon whose sides are each 102 rods? Ans. 80050.5 rods. 2. What is the area of a regular dodecagon whose sides are each 87 feet? SECTION II. THE QUADRATURE OF THE CIRCLE AND ITS PARTS. ART. 18. Definition I. A circle is a plane bounded by a line which is equally distant in all its parts from a point within called the centre. The bounding line is called the circumference or periphery. An arc is any portion of the circumference. A semi-circle is half, and a quadrant onefourth of a circle. II. A Diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. A Radius is a straight line extending from the centre to the circumference. A Chord is a straight line which joins the two extremities of an arc. III. A Circular Sector is a space contained between an arc and the two radii drawn from the extremities of the arc. * Thomson's Legendre 1, 5. Cor. VI. A Circular Ring is the space between the peripheries of two concentric circles, as AA', BB'. (Fig. 13.) VII. A Lune or Crescent is the space between two circular arcs which intersect each other, as ACBD. (Fig. 14.) 19. The Squaring of the Circle is a problem which has exercised the ingenuity of distinguished mathematicians for many centuries. The result of their efforts has been only an approximation to the value of the area. This can be car ried to a degree of exactness far beyond what is necessary for practical purposes. given diameter For the area is 20. If the circumference of a circle of were known, its area could be easily found. equal to the product of half the circumference into half the diameter. (Sup. Euc. 5, 1.*)† But the circumference of a circle has never been exactly determined. The method of approximating to it is by inscribing and circumscribing polygons, or by some process of calculation which is, in principle, the same. The perimeters of the polygons can be easily and exactly determined. That which is circumscribed is greater, and that which is inscribed is less, than the periphery of the circle; and by increasing the number of sides, the difference of the two polygons may be made less than any given quantity. (Sup. Euc. 4, 1.) 21. The side of a hexagon inscribed in a circle, as AB, is the chord of an arc of 60°, and therefore equal to the radius. (Trig. 95.) The chord of half this arc, as BO, is the side of a polygon of 12 equal sides. By repeatedly bisecting the arc, and finding the chord, we may T N B C D H F G obtain the side of a polygon of an immense number of sides. Or we may calculate the sine, which will be half the chord of double the arc, (Trig. 82, cor.,) and the tangent, which will be half the side of a similar circumscribed polygon. Thus the sine AP, is half of AB, a side of the inscribed hexagon; and the tangent NO is half of NT, a side of the circumscribed hexagon. The difference between the sine and the arc AO is less than the difference between the sine and the tangent. In the section on the computation of the canon, (Trig. 223.) by 12 successive bisections, beginning with 60 degrees, an are is obtained which is the 2476 of the whole circumference. * In this manner, the Supplement to Playfair's Euclid is referred to in this work. + Thomson's Legendre, 11. 5. The cosine of this, if radius be 1, is found to be .99999996732 The diff. between the sine and tangent is only .00000000001 And the difference between the sine and the arc is still less. Taking then, .000255663465 for the length of the arc, multiplying by 24576, and retaining 8 places of decimals, we have 6.28318531 for the whole circumference, the radius being 1. Half of this, 3.14159265 is the circumference of a circle whose radius is, and diameter 1. 22. If this be multiplied by 7, the product is 21.99+or 22 nearly. So that, Diam Circum :: 7 : 22, nearly. If 3.14159265 be multiplied by 113, the product is 354.9999+, or 355, very nearly. So that, Diam Circum:: 113: 355, very nearly. The first of these ratios was demonstrated by Archimedes. There are various methods, principally by infinite series and fluxions, by which the labor of carrying on the approximation to the periphery of a circle may be very much abridged. The calculation has been extended to nearly 150 places of decimals. But four or five places are sufficient for most practical purposes. After determining the ratio between the diameter and the circumference of a circle, the following problems are easily solved. PROBLEM I. To find the CIRCUMFERENCE of a circle from its diameter. 23. MULTIPLY THE DIAMETER BY 3.14159.* Or, Multiply the diameter by 22 and divide the product by 7. Or, multiply the diameter by 355, and divide the product by 113. (Art. 22.) Ex. 1. If the diameter of the earth be 7930 miles, what is the circumference? Ans. 249128 miles. 2. How many miles does the earth move, in revolving round the sun; supposing the orbit to be a circle whose diameter is 190 million miles? Ans. 596,902,100. 3. What is the circumference of a circle whose diameter is 769843 rods? PROBLEM II. To find the DIAMETER of a circle from its circumference. 24. DIVIDE THE CIRCUMFERENCE BY 3.14159. Or, Multiply the circumference by 7, and divide the product by 22. Or, multiply the circumference by 113, and divide the product by 355. (Art. 22.) Ex. 1. If the circumference of the sun be 2,800,000 miles, what is his diameter ? Ans. 891,267. 2. What is the diameter of a tree which is 5 feet round? 25. As multiplication is more easily performed than division, there will be an advantage in exchanging the divisor * In many cases, 3.1416 will be sufficiently accurate. |