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base. The area is then found by multiplying together the base and half the perpendicular, or, the perpendicular and half the base.
5. How many square inches in a triangle, whose base is 17} inches, and whose perpendicular height is 111 inches ?
6. How many square feet in a board 18 feet long, 16 inches wide at one end, and tapering to a point at the other end?
7. How many square feet in a plank 14 feet long, 17 inches wide at one end, and 10 inches wide at the other end ?
In this example, add the width of the two ends together, and take half the sum for one of the factors.
Area of Circles. To find the area of a circle, multiply the circumference by half the diameter, and divide the product by 2. When either the circumference, or the diameter is the only dimension known, the other dimension may be found, as stated in page
173. 8. What is the area of the head of a cask, the diameter of which is 18 inches ?
9. Suppose a cylinder to measure 3 feet in circum ference; what is the area of one end ?
AREA OF GLOBES. To find the convex area of a globe or sphere, multiply the circumference and diameter together. When the diameter is not known, it may be found from the circumference, as stated in page 173.
10. How many square inches are there on the surface of a globe, whose circumference is 14 inches ?
11. Suppose the earth to be 25020 miles in circumference, what must be the area of its whole surface ?
SOLIDS AND CAPACITIES.
It has already been taught, that solids and capacities are measured in cubes. It has also been shown, that the contents of any thing having six sides-- its opposite sides
being equal, and all its angles being right angles, are found by multiplying together the length, and breadth, and depth of the thing.
SolidiTY OF WEDGES. To find the solid contents of a wedge, first, find the area of the head or end of the wedge, and then multiply this area by half the length.
12. How many solid inches are inere in a wedge, 12 nches long, 3 inches wide, and 14 inch thick at the head?
13. What are the solid contents (in feet and inches) of a plank, 15 feet long, 17 inches wide, 24 inches thick at one end, and the thickness tapering to nothing at the other end ?
14. What are the solid contents of a stick of hewn timber, measuring in length 13 feet, in breadth 2ft. 4 in., in depth 2 feet at one end, and i st. 6 in. at the other end
In this example, add the depth of the two ends together, and take one half of the sum for the depth to be used in the multiplication.
SOLIDITY OF PRISMS. A prism is a body with two equal ends, which are either square, triangular, or polygonal, and three or more sides, which meet in parallel lines, running from the several angles of one end to those of the other. The adjoined is a representation of a triangular prism.
The solid contents of prisms of all kinds, whether square, triangular, or polygonal, are found by one general rule, viz. Find the area of the end or base, and multiply this area by the length or height.
15. How many cubic inches are there in a triangular prism, which is 16 inches in length, the ends measuring 1.2 inches on a side, and 1.01 inches perpendicular ?
16. How many cubic feet are there in a stick of timber 18 feet long, hewn 3 square, the ends forming equilateral triangles of 10 inches side, and 8.7 inches perpendizular?
Solidity of CYLINDERS. A cylinder is a round body, the two opposite sides, or ends of which, are circular planes, equal, and parallel. For instance, a stick of round timber of uniform circumference, having its ends. sawed at right angles with its length, is a cylinder: also, a common giindstone is a cylinder. To find the solid contents of a cylinder, first, find the area of one end, and then multiply this area by the length.
17. What are the solid contents of a cylinder whose length is 5 feet, and circumference 6.4 feet? (To find the diameter, see page 173.)
18. What are the contents of a cylinder whose length is 2 seet, and diameter 10 inches ?
Solidity of PYRAMIDS. Solids, which decrease gradually from the base, till they come to a point, are called pyramids. They are of different kinds, according to the figure of their bases. If the pyrainid has a square base, it is called a square pyramid; if a triangular base, a triangular pyramid; if the base be a circle, a circular pyramid, or a Cone. The point in which the pyramid ends is called the verter. A line through the centre of the pyramid, from the vertex to the base, is the height.
The Frustrum of a pyramid is what remains, after any portion of the top has been cut off, parallel to the base.
To find the cubical contents of a pyramid, first find the area of the base, then multiply this arta by one-third of the height.
19. How many cubic inches are there in a square pyramid, 3 feet in height, and 9 inches square at the
20. How many cubic inches are there in a triangular pyramid, measuring 4 feet in height, 12 inches on each side of the base, and 10.4 inches from either angle of the base perpendicular to the opposite side?
21. How many cubical inches in a cone, the height of which is 19 inches, and the diameter of the base 12 inches ?
1. SOLIDITY OF FRUSTRUMS. To find the cubical con tents of the frustrum of a square pyramid, multiply the side of the base by the side of the top, and to the product add one-third of the square of the diference of the sides, and the sun will be the mean area between the two ends. Multiply the mean area by the height, and the product will be the cubical contents k. To find the cutical contents of the frustrum of a Cone, multiply together the diameters of base and top, and to the product add one-third of the square of the difference of the diameters; then multiply this sum by .7854, and the product will be the mean area between the two ends. Multiply the mean area by the height, and the product will be the cubical contents.
22. How many cubical inches in the frustrum of a square pyramid, 20 inches in height, 12 inches square at the base, and 5 inches square at the top ? 23. How
many cubic feet in a stick of hewii timber, 18 feet long, 16 inches square at one end, and 12 inches square at the other end ?
24. How many cubic inches are there in the frustrum of a cone, measuring 3 feet in height, 16 inches in diameter at the base, and 6 inches in diameter at the top'
25. How many gallons of water can be contained in a round cistern, 6 feet in height, 4 feet in diameter at the bottom, and 3 feet in diameter at the top? (Allow 21. cubic inches to the gallon.)
SOLIDITY OF GLOBES. To find the cubical contents of a globe or sphere, first, find the convex aren, as before direcled, then multiply the area by one-sixth of the diameter; the product will be the cubical contents.
26. What are the cubical contents of a globe measuring 25 inches in circumference ?
27. How many cubic miles does the earth contain, allowing its circumference to be 25020 miles ?
SOLIDITY OF. IRREGULAR Bodies. The cubica) contents of a body, which cannot be reduced to regular geometrical form may be found as follows. Immerse it in a vessel partly full of water; then the contents of that part of the vessel filled by the rising of the water will be the contents of the body immersed.
28. How many cubic inches are there in a lobster, which, being immersed in a bucket 10 inches in diameter at top and bottom, raises the water 3 inches ?
GAUGING OF CASKS. Although the difficulty of getting the true dimensions of the interior of casks, and the variety of their curve, must prevent perfect accuracy in their mensuration, yet, by careful observation in taking the dimensions, a result may be had, which will be sufficiently correct for all common purposes.
Take the interior length of the cask, the diameter at the bung, and the diameter at the head, all in inches. Subtract the head diameter from the bung diameter, and note thë difference.
If the staves of the cask be much curved between the bung and head, multiply the difference noted by .7; if but LITTLE curved, by :6; or, if they be of a MEDIUM curve, by .65; and add the product to the head diameter; the sum is the mean diameter, and thus the cask is re. duced to a cylinder.
Square the mean diameter, and multiply the square by the length of the cask; then divide this product by 294, and the quotient will be the number of wine gallons, which The cask may contain.