RULE. Multiply the length of one of the sides by the number of sides; then, this product by the half of a perpendicular let fall from the centre of the figure to the middle of one of the sides, and the product will be the area of the polygon. In the pentagon ABCDE, each side is 95, and the perpendicular FG 65-36, to find the area. 95 length of a side. A 5 number of sides. Set 1 on A to the perpendicular on B, and against the sum of the sides on A you will have the area on B. By Gunter. The extent from 1 to half the length of the perpendicular, wilk reach from the sum of the sides to the content. But for the more ready measuring regular polygons, the following Table, containing multipliers for all regular figures from the triangle to the dodecagon, will be of use to the learner. If the square of the side of a polygon be multiplied by the multiplier of the like figure, the product will be the area of the figure sought. To measure a Circle and its Parts. In the annexed circle ABCD, the arch line ABCD is called the periphery, the length of which is called the circumference: Any line, as DB or AC, passing through the centre E, cuts the circle into two equal parts, called se- D micircles, or half circles; and such lines are called diameters of the circle: If two diameters be drawn through a circle, at right angles to each other, then, the four equal divisions of the E C circle are called quadrants: half the diameter as EB, is called the radius, or semidiameter. ART. 12. The Diameter of a Circle being given, to find the Circum ference. * RULE. This may be done by either of the following proportions in whole numbers, as 7 is to 22, or more exactly, as 113 is to 355; or in decimals, as 1 is to 3.14159; so is the diameter of a sircle to the circumference. be multiplied { divided *Note. 1. If the diameter of any circle by be be be be S multiplied 2. If the diameter of any circle {1-128379, the quotien is the side of an equal square. 3. If the diameter of any circle {-866024, the product is the side of the equilateral the quotient triangle inscribed. 4. If the diameter of any circle {1-414219, the quotient (inscribed. 707016, the product is the side of the square 5. If the square of the diameter of any circle multiplied by 1-743211, the quotient divided is the area. 6. If the circumference of any circle 31831, the product is the diameter. 7. If the circumference of any circle 2756646, the product is the side of the equilateral 3.6275939, the quotient (triangle inscribed. 9. If the circumference of any circle vitiplied by {4-442077, the quotient square inscribed. 225073, the product 2 is the side of the { multiplied by { divided (multiplied divided is the area. 10. If the square of the circumference of any circle by EXAMP. A circle whose diameter is 12, to find the circumference. As 7 22: 12 As 1: 3.14159 :: 12 12 12 As 113: 355 :: 12 12 Note. 3.14159 may be contracted to 3-1416 without any sensible difference. ART. 13. The Circumference of a Circle being given, to find the Di ameter. RULE. As 22 is to 7; or 355 to 113; or as 1 to 31831, so is the circumference of a circle to the diameter. EXAMP. The circumference of a circle being 326, to find the diameter. be { multiplied by 12. If the area of any circle 12-56636217, the product is the square of the { .079577525, the quotient (circumference. 13. When the diameter of 1 circle is 1, and the diameter of another is 2, the circumference of the first is equal to the area of the second,=3.141592. { divided 14. If the circumference be 4, the diameter and area are equal,=1.273241. 15. If the diameter be 4, the circumference and area are equal, 12.566368. Hence, because circles are the most capacious of all figures, if the fourth part fa circle be squared, it will not be equal to the area of that circle (as is commony supposed) although the four sides added together are equal to the circumfe rence of that circle. In a circle whose diameter is 24, circumference 75.4, and area 452-4, the fourth part of the circumference is 18-35, the square of which is only 355 3225, that is, 57-0775 less than the truth: and the larger the circle is, the greater will the errour be. For further proof of this matter: If a cylindrical pint, beer measure, whose content is 35.25 cubick inches, be beaten into a perfectly square form, it will contain only 28.902 cubick inches, which is less than the truth by 6-31844; the area of the circle is 8-7615853288, and the area ofthe square only 6-8313320653076624. Hence appears the reason, why taking the fourth part of the girth in measurJug a cylinder (or a round stick of timber) is false. 16. If the diameter of one circle be double to that of another, the area of the frat circle will be four times the area of the second, because the areas of circles e as the squares of their diameters; see Art. 15. ART. 14. To find the Area of a Circle. RULE. Multiply half the diameter by half the circumference and the product is the area. If the diameter be given, find the circumference by Art. 12. If the circumference be given, find the diameter by Art. 13. EXAMP. A circle whose diameter is 12, and circumference is 37-7, given, to find the area? Note. 18.85 half the circumference. 6 half the diameter. 113.10 area of the given circle. A circular ring is the figure contained between the peripheries of two concentric circles. Hence, the area of a circular ring must be the difference of the areas of the two circles. ART. 15. The Diameter being given to find the Area of a Circle without finding the Circumference. RULE. Multiply the square of the diameter by 7854,* and the product will be the area of the circle, whose diameter was given. EXAMP. The diameter of a circle being 12, to find the area? 7854 12x12 144 31416 31416 7854 113.0976 area. When the diameter is 1, the area is found to be ·7854, and as the areas of circles are as the squares of their diameters, the rule is evident. By the Sliding Rule. Set 1 on A to the diameter on B, then find 7854 (which expresses the area of a circle whose diameter is 1) on A, against which on B is a 4th number, then find this 4th number on A, against which on B is the area. By Gunter. The extent from 1 to the length of the diameter reaches from 7854 to a 4th number, and from that 4th number to the area. ART. 16. The Circumference of a Circle being given, to find the Area without finding the Diameter. RULE. Multiply the square of the circumference by 07958, and the product will be the area of the circle. EXAMP. area. The circumference of a circle being 37-7, to find the ART. 17. 1421-29 square. 113 1062582=area of the circle. The Dimensions of any of the parts of a Circle being given, to find the side of a Square equal to the Circle. RULE. If the area of the circle be given, extract the square root of the area, which will be the side of a square equal to the circle : If the diameter or circumference be given, find the area by Art. 15 or 16, and then extract the square root, as before. And this is a general rule to find the side of a square equal to any superficial fig. ure, regular or irregular: for the square root of the area of any figure whatever, is the side of a square equal to the given figure. But with regard to circles, if the diameter be given; multiply it by 886, and the product will be the side of an equal square: or, as 13-545 is to 12, or 1354 to 1200: so is the diameter of a circle to the side of a square equal to the given circle. And, if the circumference be given, multiply it by 282 for the side of an equal square. Or, divide it by 3.545, and the quotient will be the side of an equal square. Let the diameter of a circle be The circumference being 37-7 12, to find the side of a square to find the side of an equal equal to the circle? 886×12=10-632=side of the spuare. Or, a 13.545: 12: 12: 10 631 the side. square? 37 7X 282 10.631=side the square. Or, 37.7-3.545=10.634. of |