Set 1 on A to the length of the 24 192 area. By the sliding Rule. the sum of the perpendiculars on B, and opposite The extent from 1 to the sum of the perpendiculars will reach from the length of the diagonal to the area. ART. 9. To measure any irregular figure. RULE. Divide the figure into triangles, by drawing diagonals from one angle to another; then measure all the triangles by either of the rules, already taught, at Article 6 or 7, and the sum of the several areas of all the triangles will be the area of the given figure. D. B In the triangle AFB the base FA is 100, and the perpendicular Ba 49; in the triangle FBE the base BE is 92, and the perpendicular Fd 52; in the triangle EBD, the base BE is the same as before, and the perpendicular De 44; and in the triangle DCB, the base DC is 80, and the perpendicular B6 38; by which the area of each may be found by Art. 6, as follows. 50 half AF. 49 perp. aB. 2450 area of AFB. 46 half BE. 2450 46 half BE. 44 perp. Dc. 8386 area of the 184 184 2392 area of FBE. 38=perp. Bb. 2024 area of EBD. 1520=area of DCB. figure ABCDEF. In dividing any irregular figure into triangles, the triangles will be less, by two, and the diagonals less, by three, than the number of the sides of the figure. If there be a long, irregular figure like the following, the mean breadth may be found very nearly, by measuring the breadth at certain equal distan- A ces along AB, and dividing the sum of the breadths by their number. B 1 2 3 6 7 8 Let the length, AB, be 16 rods, the 1st breadth AC 39 rods, the 2d 4 rods, the 3d 395 rods, the 4th 4-3 rods, the 5th 4.25 rods, the 6th 4.5 rods, the 7th 4·8 rods, and the 8th 49 rods; what is the area? 3.9+4+395+4·3+4.25 +4·5+4·8+49 ART. 10. To measure a Trapezoid. 34.6 the mean 8 Definition. A trapezoid is the segment of a triangle, cut by a line parallel to the base. RULE. Add the parallel sides together, and multiply half that sum by the perpendicular breadth. In the trapezoid 24=AD ABCD, the side AD is 24, the side BC is 16, and the perpendicular breadth Ba is 10, to find the area by adding the sides BC and AD 16=BC 40=sum. 20 sum. and multiplying half 200=area. B their sum by the perpendicular breadth Ba. By the Sliding Rule. Set 1 on A to the equated length on B, and against the breadth on A you will have the area on B. By Gunter. The extent from 1 to the breadth will reach from the equated length to the area. ART. 11. To measure any regular Polygon. Definition. A regular polygon is a figure whose sides and angles are all equal; they are usually denominated from the number of their sides. 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 95 length of a side. area. A B. 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, will 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 A 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 Circumference.* 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 circle to the circumference. *Note. 1. If the diameter of any circle be multiplied} by be S multiplied {divided by 3.14159, the product 31831, the quotient is the circumference. 886227, the product 1-128379, the quotient is the side of an equal square. 3. If the diameter of any circle by {.866024, the product is the side of the equilateral by .1547, the quotient triangle inscribed. 4. If the diameter of any circle 707016, the product is the side of the square 1-414213, the quotient (inscribed. 5. If the square of the diameter of any circle Smultiplied by 2 divided be be{ be be 785398, the product 1-273241, the quotient is the area. multiplied by 3.14159, the quotient divided Smultiplied 2 divided by is the diameter. 7. If the circumference of any circle Smultiplied by 3-6275939, the quotient (triangle inscribed. 2756646, the product is the side of the equilateral divided 9. If the circumference of any circle Smultiplied 2 by 442877, the quotient square inscribed. 2 divided 10. If the square of the circumference of any circle { multiplied by ( { 11. If the area of any circle be{ mividplied by of 2785398, the quotient ( the diameter. EXAMP. A circle whose diameter is 12, to find the circumference. As 7 22: 12 As 1 3.14159 :: 12 12 As 113: 355 :: 12 7)264(37.71 cir-113)4260(37-699 cir. 12 37.69908 cir. 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 divided 12. If the area of any circle by 12-56636217, the product is the square of the 1-273241. 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. 14. If the circumference be 4, the diameter and area are equal, 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 of a circle be squared, it will not be equal to the area of that circle (as is commonly supposed) although the four sides added together are equal to the circumference of that circle. In a circle whose diameter is 24, circumference 75'4, and area 452·4, the fourth part of the circumference is 18.85, the square of which is only 355-3225, that is, 97-0775 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 6.3484+; the area of the circle is 8-7615859288, and the area of the square only 6-8813320653076624. Hence appears the reason, why taking the fourth part of the girth in measuring 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 first circle will be four times the area of the second, because the areas of circles are as the squares of their diameters; see Art. 15. |