To gauge a dodecahedron :-Place twice the distance between two parallel planes over the gauge points for the pentagon, or for the pentagonal prism, and over the edge or side found on D, will be found the cubic contents on the line C. The gauge points are (for feet, when the dimensions are taken in feet) 0.7623; and 9.15, (for feet, when the distance between the parallel planes is taken in feet, and the side in inches ;) and 10.205 for bushels, (when the distance between the parallel planes is taken in feet, and the side in inches.) To find the solidity of an icosahedron : Multiply its surface by one-sixth of the distance between the parallel faces. Or, Multiply the cube of one of its edges by 2.181695, and the product will be the solidity. To find the surface of the icosahedron :— Multiply the square of one of its edges by 8.6602540. To find the perpendicular distance between two parallel faces of the icosahedron : Multiply the side, or one of the edges, by 1.5115226. To gauge an icosahedron :-Place ten-thirds of the distance between the parallel planes, over the gauge points for the equilateral triangular pyramid, and over the side found on D, will be found the contents on C. Or :-Place the distance in feet between two parallel surfaces over 11.143 on D, and over the side in inches will be found the contents in bushels; or, over 9.99, and over the side in inches will be found the contents in cubic feet; or, if we place the distance between the parallel surfaces in inches, over 13.98 on D, over the side in inches found on D will be found the contents in ale gallons; or, over 12.65, and over the side in inches found on D, will be found the contents in wine gallons. EXAMPLES. 1. What is the solidity of a dodecahedron the side of which is 6? Ans. 1655.23. 2. What are the solidity and surface of a dodecahedron whose side is 4? Answers, 490.4396, and 330.332. 3. How many feet and how many bushels are contained in a dodecahedron, the distance between the parallel faces being 5 feet, and the side 26.94 inches? Answers, 70 bushels, and 87 feet, nearly. 4. Find the solidity of an icosahedron whose edge is 6. Ans. 471.245. 5. What are the solidity and surface of an icosahedron, whose edge is 5? Ans. 272.71187, and 216.50635. 6. How many solid feet, ale and wine gallons, and bushels, in an icosahedron, the distance between its parallel faces being 24 inches, and its side 16.54 inches? Ans. 41 wine, 33.62 ale gallons; 4.405 bushels; and 5.485 feet. 7. What is the solidity of a hexahedron, of a tetrahedron, of an octahedron, of a dodecahedron, and of an icosahedron, the edge of each being 6 inches? Answers, 216; 25.4559; 101.823372; 1655.2336; and 471.24612 cubic inches. 8. How many times greater is a dodecahedron 12 inches on a side, than one 6 inches on a side? Ans. 8. The following table exhibits the surfaces and solidities of the regular solids whose edges are 1 :— To find the solidity of the frustum of a regular pyramid :Complete the solid; and then find the solidity of the whole pyramid, abcde, and also the solidity of the pyramid, ahigs, standing on the less base; then subtract the contents of the pyramid, ahigs, from the solidity of the whole pyramid, abcde, and the remainder will be the solidity of the frustum. To find the altitude of the pyramid when completed, say:As the difference of the edges of the less and greater base is to the greater base, so is the perpendicular altitude of the frustum to the required altitude of the pyramid when completed. Or, To find the solidity of the frustum of a square pyramid :— Multiply the edge of the greater base by the edge of the less, and to the product add one-third of the square of the difference of the edges of the less and greater base, and the sum will be the mean area, which, being multiplied by the altitude of the frus tum, will give its required solidity.-Apply this rule to the frustum of the equilateral triangular pyramid, and then multiply the square mean area by 0.4330127, and the product will be the required mean area of the triangular frustum, which, multiplied by its altitude, will give the required solidity. The same rule will apply to the frustums of the regular pentagonal, hexagonal, heptagonal pyramids, &c., if we multiply the square mean areas by the tabular areas of the regular polygons, whose edge is 1. See 21, page 63. The solidities of similar cones, or of similar pyramids, are to each other as the cubes of their bases, or as the cubes of their altitudes. Thus, in figure 2, the solidity of the triangular pyramid, abc, is to the cube of its base, bc, as the solidity of the pyramid, adg, is to the cube of its base, dg. Or, the solidity of the former pyramid is to that of the latter, as the cube of the base of the former to that of the latter; and the same is true of the cubes of their altitudes. To find the solidity of the frustum of any pyramid : Multiply the area of the greater base by one of its edges, and the area of the less base by its corresponding edge; divide the difference of the products by the difference of these edges, and the quotient divided by 3 will give the mean area, which, multiplied by the altitude of the frustum, will give the solidity. EXAMPLES. 1. Find the solidity of a frustum of a of the ends, or bases, being 3 feet and 2 height 5 feet. pyramid, the edges feet square, and its Ans. 3711 feet. 2. Find the solidity of a frustum of a square pyramid, the edges of its ends being 10 and 16 inches, and its length 18 feet. Ans. 21 feet. 3. Find the solidity of a frustum of a regular hexagonal pyramid, the edges of its ends being 4 and 6 feet, and its length 24 feet. Ans. 1579.6302 feet. 4. Find the solidity of a frustum of a regular octagonal pyramid, the edges of its bases being 3 and 5 feet, and its height 10 feet. Ans. 788.648 feet. 5. What is the solidity of the frustum of an equilateral triangular pyramid, the sides of the greater base being 15, and of the less 6, and the height 40? Ans. 2026.44. 6. The altitude of a square pyramid is 663 feet, and its base 15 feet on a side; it is required to find the distance from the vertex of the pyramid of a plane, which will cut off onefifth of the solid contents. Ans. 38.9863 feet. 7. The altitude of an equilateral triangular pyramid is 9, and the side of its base 6; at what distance above the base must it be cut off, in order to divide it into two equal parts? Ans. 1.80165. To find the surface of a frustum of a pyramid : When the pyramid is regular, multiply the sum of the perimeters of the two ends by the lateral length, and to half the product add the areas of the two ends, and the sum will be the surface. When the pyramid is irregular, the lateral planes are trapezoids, and their areas may be found separately by the rule for the trapezoid, ¶ 18, and to their sum add the areas of the two ends for the whole surface. 8. Find the surface of the frustum of a square pyramid, the sides of its ends being 14 and 24 inches, and the lateral length 2 feet 3 inches. Ans. 19.61 feet. 9. What is the surface of the frustum of a regular pentagonal pyramid, its lateral length being 5 feet 10 inches, and the sides of its ends 10 and 15 inches? Ans. 34.2649 feet. 10. Find the solidity of the frustum of a regular pyramid of ten sides, the edges of its ends being 4 and 6 feet, and its length 30 feet. Ans. 5847.597 cubic feet. 12 |