EXAMP. 2. In a quadrangular pyramid, the height BE being 48, and each side of the base 13, to find the content. EXAMP. 3. To measure a Cone.-The diameter AC heing 13, and the height BD 48, to find the 87 E D D the 7963956 1327326 2123-7216= content. Note The superficial content of all pyramids is found by taking of the several areas, which compose them. That of a by multiplying the circumference of the base into half the tine joining the vertex and any point in that circumference, and adding the area of the base to the product, ART, ART. 33. To measure the Frustum of a Pyramid. Definition. The frustum of a pyramid is what remains after the top is cut off by a plane parallel to the base, and is in the form of a log greater at one end than the other, whether round, or hewn three or four square, &c. RULE. If it be the frustum of a square pyramid, multiply the side of the greater base by the side of the less; to this product add one third of the square of the difference of the sides, and the sum will be the mean area between the bases; but if the base be any other regular figure, multiply this sum by the proper multiplier of its figure in the Table, Art. 11. and the product will be the mean area between the bases: lastly, multiply this by the height, and it will give the height of the frustum. EXAMP. J. In the frustum of a square pyramid the side of the greater base AD less, BC = 6, and the height EF content. = 15, the side of the F Or, if it be a tapering square stick of timber, take the girth of it in the middle; square of the girth (or multiply it by itself in inches) then say, as 144 (inches) to tha product; so is the length, taken in feet, to the content in feet. EXAMP. 2. What is the content of a tapering square stick of timber, whose side of the largest end is 12 inches, of the least end, 8, and whose length is thirty feet. One fourth of the girth in the middle = 10, and 10x10 = 100, the area in the middle; then, as 144: 100 :: 30 feet: 20.83 feet the content. By the Sliding Rule. Set 12 on D to of the circumference on C, and against the length on D is the answer on C. By Gunter. The extent from 12 to 4 of the circumference doubled, or twice turned over, will reach from the length to the content. EXAMP. 3. In the frustum of a triangular pyramid, F the side of the greater base AC 15, as before, the BD side of the less BD: : 6, and the height EF = 40, to find the content. 15 AC. = 6 = BD, 19= difference of the sides. 9 15 6 90 Add 27 Or, if it be a tapering three square stick of timber, you may find the area midway from end to end, then, as 144 is to that area, so is the length, taken in feet, to the content in feet. EXAMP. 4. To measure the Frustum of a Cone. RULE.-Multiply the diameters of the two bases together, and to the product add one third of the square of the difference of the diameters: then multiplying this sum by 7354, it will be the mean area between the two bases, which being multiplied by the length of the frustum, will give the solid content. Or, to the areas of the top and bottom add the square root of the product of those areas, and the sum, multiplied by one third of the height of the frustum, will give the solidity. When figures run uniformly taper; but not to a point (they being considered as portions of the cone or pyramid) we may find the solidity by supplying what is wanting to complete the figure, and then deducting the part cut off. A general rule for completing every straight sided solid, whose ends are paral lel and similar. As the difference of the top and bottom diameters is to the perpendicular height, (or depth which is the same :) so is the longest diameter to the altitude of the whole cone or pyramid. The The former cone in Art. 32, Examp. 3, being cut off in the middle, the greater diameter AC is 13, the less BD 64, and height EF 24, to find the content of the frustum. Definition. A sphere or globe is a round solid body, in the middle of which is a point, from which all lines drawn to the surface are equal. RULE. Multiply the cube of the diameter by 5236, and the product will be the solid content. Or, multiply the circumference by the diameter, which will give the superficial content; then multiply the surface by one sixth of the diameter, and it will give the solidity. Or, multiply the cube of the diameter by 11, and the product divided by 21, will give the solidity. EXAMP. The diameter, AB, of a globe, is 4-5 feet; to find the solid content. Note. If the circumference, or greatest circle of the sphere, be giv en, multiply the cube of it by 016887 for the content. The surface of the globe may be found by multiplying the squareof the diameter by 3-1416; or by multiplying the area of its greatest circle by 4, or the square of the circumference by ⚫3183. When the solidity of a globe is given, the diameter may be found by dividing the solidity by 5236, and extracting the cube root of the quotient. Or. if the circumference be required, divide the solidity by 016887, and the cube root of the quotient will give it. ART. 35. To measure the Solidity of a Frustum or Segment of a Globe. Definition. The frustrum of a globe is any part cut off by a plane. RULE. To three times the square of the semidiameter of the base, add the square of the height; then multiplying that sum by the height, and the product by 5236, you will have the solid content. EXAMP. The height BD being 9 inches, and the diameter of the base AC 24 inches: to find the content. |