The following TABLES of Areas and Solidities will be found considerably to diminish the labour of calculation. The numbers in the first column after the names represent the areas or solidities, when the length of the side or edge is 1, or unity; and the numbers in the other columns are multiples of those in the first, by the unit over each. To find, therefore, the area of a Polygon, take the square of the length of the side, and seek in the proper columns the multipliers, which are to be ranged under each other as in common multiplication. The sum of these will be the area required. EXAMPLE. To find the area of a Pentagon, whose side is 18 inches, the multiplier being the square of 18 = 324. The number in the Table under 4 is 66 66 66 ،، 66 2 " 66 66 66 Answer, the last four figures being decimals 66 6.8819 3.4409 66 35.1614 (the Area,) 5 5 7·4 3 0 9. The same RULE is applicable to Table X., of the Solidities of Regular Bodies, using the cube instead of the square. To find the Areas of Regular Polygons. RULE.-Square the length of side of the Polygon, and take the products from the subjoined Table, as directed in page 66. 1 2 3 4 5 6 7 8 •4330 3.0311 ⚫8660 1.2990 1.7320 2.1651 3.4641 1.7204 12.0433 12.9903 2.5981 6.1818 12.363618-5455 24-7273 To find the Areas of Circles and Spheres. RULE.-Square the diameter, and take the products from the subjoined Table, as directed in page 66. 1 Area of Circle... .7854 8-1416 Area of Sphere..] 3·1416 Circumference of Circle... 9 3.8971 23.3826 29.0713 | 32.7052 56.1938 65.5595 74.9251 84.2908 67.1769 78.3731 89.5692 100.7654 2 1.5708 6.2832 6 7 3 4 2.3562 3.1416 3.9270 4.7124 5.4978 | 9.4248 12.5664 15.7080 18.8496 21.9912 | 9.4248 12.5664 15.7080 18.8496 21.9912 5 8 9 6.2832 7.0686 25.1328 28.2744 To find the Areas of the Regular Bodies. RULE.-Square the length of one of the edges, and take the products from the subjoined Table, as directed in page 66. .... Tetrahedron 1.7320 2 3 4 6 7 8 9 8.6603 10-3923 12.1244 13.8564 15.5885 3.4641 5.1962 6.9282 12.0000 18.0000 | 24-0000 30.0000 36.0000 | 42.0000 48.0000 54.0000 6.9282 10.3923 13.8564 17-3205| 20-7846 24.2487 27-7128 41.2915 61.9372 82-5829 103.2287 123-8744 144-5201 165.1658 185.8116 | | 8.6603 17.3205 25.9808 34-6410 43.3013 51.9615 60-6218 31.1769 69.2820 77.9423 Tetrahedron Octahedron 1 To find the Solidities of the Regular Bodies. RULE.-Cube the length of one of the edges, and take the products from the subjoined Table, as directed in page 66. 5 2 •2357 2.0000 .9428 3 1 4 •1178 .3536 .4714 1.4142 1.8856 6 •5893 •7071 53.6418 5 7 TABLE of Squares, Cubes, Square Roots, and Cube Roots. Square Root. 1.0 1.4142136 2.0 2.2360680 2.4494897 2.6457513 2.8284271 3.0 3.1622777 3.3166248 2.2239801 11 3.4641016 2.2894286 12 3.6055513 2.3513347 13 3.7416574 2.4101422 14 3.8729833 2.4662121 15 4.0 2-5198421 16 2.5712816 17 2-6207414 18 2-6684016 19 4.1231056 4.2426407 4.4721360 2.7144177 20 54872 59319 64000 5.0990195 5.1961524 5.2915026 Cube Root. Number. 5.3851648 5.4772256 1.0 1.8171206 1.9129312 2.0 2.0800837 2.1544347 12345 6 7 8 9 10 4.5825757 2-7589243 21 4.6904158 2.8020393 4.7958315 2.8438670 23 4.8989795 2.8844991 24 5.0 2-9240177 25 2.9624960 26 3.0 3.0365889 3.0723168 3.1072325 30 22 22 29791 5.5677644 32768 5.6568542 3.1413806 31 3.1748021 32 3.2075343 33 35937 5.7445626 39304 5:8309519 3.2396118 34 5.9160798 3.2710663 35 42875 6.0 3.3019272 36 6.0827625 3.3322218 37 6.1644140 3.3619754 38 6.2449980 3.3912114 39 6.3245553 3.4199519 40 69 |
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