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Art. 223. TO FIND THE SOLID CONTENT OF A FRUSTUM OF A

PYRAMID.

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; 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. 200, and the product will be the mean area between the bases; then multiply this mean area by the perpendicular height of the frustum, the product will be the solid content.

1. Suppose the perpendicular height of the triangular frustum E to be 12 feet, the side of the base at the bottom 4 feet, the side at the top 2 feet; what is its solid content?

4X2=8+1.3339.333 X .433=4.041189 square feet, the mean area. And 4.041189 X 12=48.494268 cubic feet, the solid

content.

2. What is the number of cubic feet in the frustum of a square pyramid, the perpendicular height of which is 36 feet, the side at the base 12 feet, and the side at the top 6 feet?

Art. 224. TO FIND THE SOLID CONTENT OF THE FRUSTUM OF A

CONE.

RULE. Multiply the diameter of the base by the diameter of the top; to the product add one third of the square of the difference of the diame ters; then multiply the sum by .7854; the product will be the mean area between the two ends. Multiply the mean area by the perpendicular height; the product will be the solid content.

1. Suppose the frustum of a cone F to be 15 feet in height, 6 feet in diameter at the base, and 3 feet in diameter at the top, what is its solidity?

=

6X3 18+3=21 X .7854 = 16.4934 square feet, the mean area; and 16.4934 X 15-247.401 cubic feet, the required solidity. 2. What is the number of cubic feet in a mast, 60 feet in length, 3 feet in diameter at the base, and 1.5 feet in diameter at the top? Art. 225. TO FIND THE AREA OR CONVEX SURFACE OF A SPHERE

OR GLOBE.

RULE. Multiply the circumference by the diameter.

[graphic]

bas

1. Suppose the diameter A B of the globe in the margin to be 12 feet, what is its convex area or surface?

12 X 3.141637.6992 X 12=452.3904 square feet, the surface required.

2. What is the surface of a sphere whose diameter is 7912 miles?

Art. 226. TO FIND THE SOLIDITY OF A SPHERE OR GLOBE.

RULE. Multiply the surface by one sixth of the diameter; the product will be the solidity.

Or, multiply the cube of the diameter by .5326; the product will be the solid content.

1. Suppose the diameter A C of the globe in the margin to be 9 feet, what is its solidity?

9 × 9 × 9=729 X.5236=381.7044 cubic feet, the required solidity.

2. What is the solid content of the globe on which we live, supposing the diameter to be 7912 miles?

Art. 227. TO FIND THE SOLIDITY OF A WEDGE.

RULE. To twice the length of the base add the length of the edge; multiply the sum by the breadth of the base, and the product by one sixth of the perpendicular from the edge upon the base; this product will be the solidity.

1. What is the number of cubic inches in a wedge, of which the perpendicular height is 8.7 inches, the edge 3.5 inches, and the base or head 3 inches in length and 2 inches in breadth?

Ans. 37.55 cubic inches.

Art. 228. THE FIVE REGULAR BODIES.

A REGULAR BODY is a solid bounded by similar and regular plane figures. There are five such regular solid bodies, viz. the Tetraedron, the Hexaedron, the Octaedron, the Dodecaedron, and the Icosaedron.

The Tetraedron, or equilateral pyramid, is a solid bounded by four equilateral triangles.

The Hexaedron, or cube, is a solid bounded by six equal squares. The Octaedron is a solid bounded by eight equilateral triangles. The Dodecaedron is a solid bounded by twelve regular and equal pentagons.

The Icosaedron is a solid bounded by twenty equilateral triangles. The following table shows the surface and solidity of each of the five regular solid bodies, the linear edge of each being unity or 1.

[blocks in formation]

Similar surfaces and solids are those which have their corresponding dimensions proportional. Since the areas of similar surfaces are to each other as the squares of their corresponding dimensions, we have the following rule for finding the surface of any regular solid, when the length of its linear edge is given.

RULE. Multiply the tabular number in the column of surfaces by the square of its linear edge; the product will be the surface required. 1. The linear edge of an octaedron is 6 feet; what is its surface? The tabular area is 3.4641016. 6 X 6 36. And 3.4641016 X

36=124.7076576 square feet, the surface required.

Since the contents of similar solids are to each other as the cubes of their corresponding dimensions, we have the following rule for finding the content of any regular solid, when the length of its linear edge is given.

RULE. Multiply the tabular number in the column of solidities by the cube of its linear edge; the product will be the solidity required. 2. What is the solidity of a regular icosaedron, whose linear edge is 5 feet?

5 X5 X 5=125. And 2.181695 X 125=272.711875 cubic feet, the required solidity.

3. What is the solidity of a regular octaedron, whose linear edge is 6 feet?

PRACTICAL QUESTIONS.

Art. 229. 1. What is the solidity of an octangular prism, whose side measures 6 inches, and whose length is 6 feet? What is the area of its surface?

2. There is a cylindrical granite pillar, 32 feet in length, 6 feet in diameter at the base, and 4 feet in diameter at the top. What is the area of its surface in square feet? What is its solidity in cubic feet?

3. What is the content of a cylindrical cistern in imperial gallons, whose depth is 6 feet, inside diameter at bottom 5 feet 9 inches, at top 5 feet 3 inches?

4. What is the content of a square pyramid, 750 feet in perpendicular height, and each side of its base measuring 620 feet?

5. What number of bushels of grain will a cubical bin contain, whose length inside is 8 feet, breadth 5 feet, and depth 4 feet?

6. There is a globe composed of India rubber cloth, 3 feet in diameter; what number of square yards of cloth were required to make the globe, making no allowance for waste; and what is its content in imperial gallons?

MENSURATION OF BOARDS AND TIMBER.

Art. 230. The unit of measure for boards, plank and joist, is the square foot.

TO FIND THE NUMBER OF SQUARE FEET IN A BOARD.

RULE. Multiply the length in feet by the width in inches; divide the product by 12; the quotient will be the square feet. If the board is tapering, add the width of the two ends, and half the sum is the mean width.

1. What is the number of square feet in a board, 16 feet in length, 15 inches wide at one end, and 13 inches at the other?

15+13=28214 in., the mean width; and 16 × 14=224 12183 square feet.

TO FIND THE NUMBER OF FEET, BOARD MEASURE, IN A PLANK OR JOIST.

RULE. Multiply the length in feet by the width in inches, and the product by the depth in incres; divide this product by 12; the quotient will be square feet.

2. What number of square feet in a plank, 12 feet in length, 15 inches in width, and 2 inches in thickness? Ans. 30 square feet.

3. What number of square feet in a joist, 16 feet in length, 3 inches in width or thickness, and 9 inches in depth?

Art. 231. TO FIND THE SOLIDITY OF SQUARE OR RECTANGULAR

TIMBER WHICH DOES NOT TAPER.

RULE. Find the area of one end by multiplying the width by the depth; then multiply this area by the length; the product will be the solidity.

1. What is the solidity of a stick of timber, 16 feet in length, 15 inches in width, and 12 inches in depth? Ans. 20 cubic feet. 2. What is the number of cubic feet in a log of mahogany, 12 feet in length, 30 inches in width, and 27 inches in depth?

Art. 232. TO FIND THE CONTENT OF ANY SOLID, OF WHICH THE BASES ARE PARALLEL, AND THE GREATEST AND LEAST THICKNESSES OR DIAMETERS ARE AT ITS ENDS.

RULE. Find the area of each end, also the mean area between the two ends; multiply the mean area by 4; to the product add the area of each end, and multiply the sum by one sixth of the length; this product will be the solid content.

NOTE 1. When the sides of the solid are straight between the ends, half the sum of two corresponding sides or diameters will be the mean corresponding side or diameter.

1. What number of cubic feet in a stick of timber 24 feet in length, the ends of which are 24 inches by 21, and 18 inches by 15?

Ans. 63.5 cubic feet.

2. What is the content of a pine log which is 18 feet in length, 21 inches in diameter at the larger end, and 18 inches in diameter at the smaller?

Art. 233. TO FIND THE SIDE OF THE LARGEST SQUARE STICK OF

TIMBER THAT CAN BE HEWN OR SAWN FROM A ROUND LOG.

RULE. Multiply the diameter of the smaller end of the log by .7071, the product will be the side of the inscribed square or the required side. 1. What will be the side of the largest stick of square timber which can be hewn from a round log, 18 inches in diameter at the smaller end? 18 X.7071 12.7278 inches.

=

2. What will be the side of the largest stick of square timber that can be sawn from a round log 12 inches in diameter at the smaller end? Art. 234. TO FIND THE NUMBER OF SQUARE FEET OF BOARDS

WHICH CAN BE SAWN FROM A LOG OF ANY GIVEN DIMENSIONS.

RULE. Find the side of the largest square stick of timber which can be sawn from the given log, Art. 233. Divide the side by the thickness of the board, plus the thickness of the saw; the quotient will be the number of boards. Multiply the number of square feet in one board by the number of boards, the product will be the required number of square feet.

1. What is the number of feet of boards, 1.25 inches in thickness, including the thickness of the saw, which can be sawn from a log 12 feet in length, and 14.25 inches in diameter at the smaller end?

=

14.25 X .7071=10.076+ inches, the side of the inscribed square; 10.0761.25 8+, the number of boards 10 inches in width, and 12 X 10=120 ÷ 12-10, the number of square feet in one board, and 10 X 880, the number of feet required.

2. What is the number of feet of boards, 1.25 inches in thickness, which can be sawn from a log 16.75 inches in diameter at the smaller end, and 16 feet in length?

MENSURATION OF MASONRY.

Art. 235. Masonry includes all kinds of work of which stone and brick are the principal materials.

The measuring unit for walls of stone or brick, columns, blocks of stone or marble, is the cubic foot.

The measuring unit for pavements, slabs, chimney pieces, &c., is the square foot.

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