In some states a standard board is 12 feet long, 12 inches wide, and 1 inch thick; and this is what is generally to be understood by a standard board. 38. PYRAMIDS. A pyramid is a solid having any rectilineal figure for its base, and its other sides are triangles, whose vertexes all meet in one point, called the vertex of the pyramid. The altitude of a pyramid is a perpendicular from its vertex on the plane of the base. The pyramid is said to be triangular, quadrilateral, polygonal, &c., according as its base is a triangle, a quadrilateral, a polygon, &c. When the base of the pyramid is regular, a right line joining its centre and the vertex is called the axis of the pyramid; and when the axis is perpendicular to the base it is called a right pyramid; but if it be not perpendicular to the base, the pyramid is said to be oblique. The pyramid is just one-third of its least circumscribing prism. Therefore, To find the solidity of a pyramid : Multiply the area of its base by one-third of its perpendicular altitude, and the product will be the solidity. To find the surface of a pyramid : If it be a right pyramid, multiply the perimeter of the base by the altitude, or apothegm, of one of its slanting sides, and add half the product to the area of the base; but if the pyramid be oblique, find the areas of the lateral triangles separately, and to their sum add the area of the base. To find the altitude of a pyramid, its base and solidity being given:-Divide three times its solidity by the area of its base, and the quotient will be the altitude. Or, three times the solidity divided by the altitude of a pyramid, will give the area of its base. If we take two square pyramids, whose bases are equal to each other, and the sides of the base equal to the lateral edges, and place the base of one of the pyramids upon that of the other, a solid will thus be formed, which is comprehended under eight equilateral triangular faces. This solid is called the OCTAHEDRON. To find the solidity of an octahedron :-Multiply the square of one of its edges by one-third of the distance between two opposite vertexes. Or, multiply the cube of one of its edges by 0.4714045 for the solidity; and multiply the square of one of its edges by 3.4641016 for the surface of its eight faces. To find the distance between the vertexes of an octahedron :Extract the square root of twice the square of one of the edges. When the base of a triangular pyramid is equilateral, and the lateral faces are likewise equilateral and equal to the base, the pyramid is called a TETRAHEdron. To find the solidity of a tetrahedron:-Multiply the cube of one of its edges by 0.1178513; and multiply the square of one of its edges by 1.7320508 for the surface of its four faces. The distance of the centre of the base of the tetrahedron from one of the angles of the base may be found by extracting the square root of one-third of the square of one of the sides of the base; and the square root of two-thirds of the square of one of the sides will be the altitude of the tetrahedron. EXAMPLES. 1. What is the solidity of a square pyramid, each side of its base being 3 feet, and its altitude 10 feet? Ans. 30 feet. 2. What is the solidity of an equilateral triangular pyramid, each side of the base being 3 feet, and its altitude 30 feet? Ans. 38.97114 feet. 3. Find the solidity of a square pyramid, each side of the base being 30 feet, and its apothegm (that is, the perpendicular height of one of its lateral triangles) 25 feet. Ans. 6,000 feet. 4. What is the solidity of a pentagonal pyramid, each side of the base being 4 feet, and its altitude 30 feet? Ans. 275.276 feet. 5. How many solid feet, and how many bushels in a pyramid, its altitude being 9 feet, and its base being 4 feet square? Answers, 48 feet, and 38.6 bushels. The gauge points for the pyramids are the same as for the corresponding prisms. Therefore, place one-third of the altitude of any pyramid over the gauge point for the prism, whose base is similar to that of the pyramid, and over the side found on D will be found the contents of the pyramid on C. See 33, under example 3d. 6. What is the content of an hexagonal pyramid, the altitude being 6.4 feet, and each side of the base 6 inches? Ans. 1.3856072 feet. 7. The hopper of a cider-mill is a square pyramid, its base being 4 feet on a side; required its length in order that it may hold 50 bushels. Ans. 11.66 feet. Place three times the contents, viz. 150 bushels, over one side of the base, reduced to inches, viz. 48, and over the gauge point for bushels in a square prism, viz. 13.387, will be found the altitude of the hopper. 8. Find the surface of a square pyramid, its apothegm being 40 feet, and each side of the base 6 feet. Ans. 516 square feet. 9. Find the surface of a pyramid whose apothegm is 10 feet, and its base an equilateral triangle, whose side is 18 inches. Ans. 23.474278 feet. 10. The apothegm of a regular hexagonal pyramid is 8 feet, and the side of its base 2 feet; what is its surface? Ans. 76.23795 feet. 11. What is the solidity of a tetrahedron, whose edge is 8? Ans. 60.33. 12. What are the solidity and surface of a tetrahedron, whose edge is 6? Answers, 25.45, and 62.35. 13. Find the solidity of an octahedron, whose edge is 16. Ans. 1930.87. 14. What are the solidity and surface of an octahedron, Answers, 12.728 and 31.177. whose edge is 3? 15. What is the altitude of a tetrahedron, whose edge is 6? Ans. 4.9, nearly. 16. What is the diameter of a globe from which you may cut an octahedron whose side is 6 inches? Ans. 8.48528 inches. 39. REGULAR SOLIDS. There are five regular solids, or, as they are sometimes called, Platonic Bodies, and it can be proved that no more can exist. These solids are the tetrahedron, hexahedron, or cube, the octahedron, dodecahedron, and the icosahedron. The first three have already been described. The dodecahedron is a solid comprehended under twelve regular pentagons; and it may be considered as a solid composed of twelve pentagonal pyramids, whose bases form the surface of the solid, and whose vertexes all meet in its centre. The icosahedron is a solid comprehended under twenty equilateral triangular faces; and the solid may be conceived to consist of twenty equal triangular pyramids, whose bases form the surface of the solid, and whose vertexes all meet in the centre. Each side of a regular solid, except the tetrahedron, has an opposite face parallel to it, and the edges of these faces are also respectively parallel. To find the solidity of a dodecahedron:— Multiply its surface by one-sixth of the distance between two parallel sides. Or, Multiply the cube of one of its edges by 7.6631189. To find the surface of a dodecahedron: Multiply the square of one of its edges by 20.6457088. To find the perpendicular distance between two parallel planes of the dodecahedron:-Multiply one of the edges by 2.2270274. |