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Practical applications of the principles of geometry
to determine the numerical relations of surfaces
bounded by straight lines. In the comparison of lines of different lengths, we employ a certain measure which is called a unit, as, for example, an inch, a foot, a yard, &c. In comparing the surfaces of figures we employ a square
described on the lineal unit of measure, which is called a square inch, a square foot, a square yard, &c. The mensuration of surfaces consists in ascertaining how many squares of a certain size are contained in the surface of a given figure. The area of a figure is the number of squares of a certain unit which it contains. How do you find the area of a square, a rectangle, a rhombus, or a rhomboid?
The base x height What is the area of a square of which the side is 21 feet? 21 x 21=441 square feet. Required the area of a rhomboid whose length is 20:5, and breadth 11•75. 20.5 x 11.75=240.875 the area. Let the sides of a rectangle be 12 and 9, what is its area? 12x9=108 the area. How do you find the area of a triangle ? Multiply the base by the perpendicular height, and half the product will be the area. Find the area of a triangle, of which the base is 115, and perpendicular 80. 115 X 40 = 4600. The base of a triangle is 76.5 feet, and per. pendicular 92.2 feet, what is its area? 76.5 x 92.2 • 2 = 3526.65 square feet, the area. Each side of a triangular field measures 4 perches, what is its area ? 42 = 16; then 16 ; 4= 4, and 16 — 4= 12; then
16 12 x = 12 x 4 = 48; N 48 = 6.928 the area.
4 Given the area of a triangle = 12 yards, and altitude = 4, what is its base? 124=3: then 3 X 2 = 6 yards the base. Given the area of a triangle
= 12 and its base =
6, what is its perpendicular height? 12 = 6 = 2; then 2 x 2 = 4 the altitude. Given the base 10, the perpendicular 12, required the length of the hypotenuse. 16+ 122 = 20, the length of the hypotenuse. How do you find the area of a trapezoid ? Multiply half the sum of the parallel sides by the perpendicular distance between them, and the product will be the area of the trapezoid. What is the area of a trapezoid, its parallel sides being 25 and 33 and the perpendicular distance between them 12 yards? } (25 + 33) x 12 = 29 X 12 = 348 square yards. How do you find the area of a circle ? 1st, diameter X 3.1416 = circumference. 2nd,
3:1416 -- diameter. 3rd, į circumference X radius = area. In a circle whose diameter is 14 inches, we have, 1st, 14 x 3.1416 = 43.9824, the circumference. 2nd, 43.9824
14, the diameter. 3rd, diameter • 2 3.1416
14 = radius, so - 7 = radius. Then _ (43.9824) X 7
2 = 153.9384, the area. The diameter of a circle is 20, what is its circumference ? 3.1416 X 20 = 62.832. The circumference of a circle is 62.832, what is the diameter ? 62.832 ; 3.1416 : 20, or 62.832 X 31831 = 20 as before. The area of a circle is to the square of the diameter as •7854 to 1; or multiply half the circumference by half the diameter and the product will be the area. Required the area of a circle the diameter being 30.5. 30:52 •7854 = 730:618350, the area required. What is the area of a circle when the diameter is 1 ? In this case the circumference is 3.1416, half of which is 1.5708, and half of 1= 3; then 1.5708 x 5 = .7854 the area.
On the most essential properties of the simpler solids
bounded by plane surfaces. A solid is a figure bounded by surfaces. The simplest solid is a cube, or figure bounded by six equal squares, and is therefore chosen as the measuring unit of solids. How do you find the solidity of a cube? Multiply the side of a cube by itself, and that product again by the side, for the solidity required. If the side of a cube be 4 inches, required its solidity. Here, 4 X4= 16, the number of cubes of 1 inch deep in the square, and as the entire solid consists of four such dimensions, its contents is 16 X 4 = 64 cubic inches. How do you
find the surface of a cube? Square the side, and multiply this square by 6, the product will be its surface. Let A be the side, then 6 A2 is the surface. The reason of this rule is obvious, for A2 is the area of one of the faces, therefore 6 Ao will be the area of the six faces. What is the surface of a cube whose side is 1 ? Here 6 A2 = 6 x 12 = 6, the surface required. What is the surface of a cube, the length of its side being 20 inches ? 202 X 6 = 2400 square inches. What is the surface of a parallelopiped of which the length is 50 feet, breadth 6 feet, and depth 4 feet? (6 + 4) X 2 X 50 = 1000 = area of the sides.
6 X 4 X 2 = 48 = area of the ends. Area of the prism = 1048 square feet. How do you find the solidity of a parallelopipedon ? Multiply continually the length, breadth, and depth together, for the solidity. What is the solidity of a parallelopipedon, the length being 10 feet, the breadth 4 feet, and depth 5 feet? 10 X 4 X 5 = 200 feet. How do you find the surface of a pyramid or cone? } (slant height x perimeter of base) + area of the base base 10 feet? 15 x 5 = 75 square feet. What is the surface of a square pyramid, its slant height being 30 feet, and each side of the base 4 feet? 4 X 4 X 15 + 4 x 4= 256 square feet. How do you find the surface and solidity of a pyramid or cone ? } (area of base x perpend. height) = content. Į (slant height X perimeter of base) + area of base = surface. Slant height is 10, perimeter of base 16; then +(10 x16)=80+16=96, surface of a four-sided pyramid, whose side at the base is 4. The area of the base of a cone being 147.68 and perpendicular height 14. Then ţ (14 x 147.68) = 689.17 content. What is the solidity of a square pyramid, each side of its base being 4 feet, and height 12 feet?
surface. What is the convex surface of a cone, its slant height being 15 feet, and the circumference of its
12 4 X 4= 16 = the area of the base. Then 16 X
3 64 feet, the solidity. How do you find the surface of a sphere or of any segment or zone of a sphere ? Multiply the square of the diameter by 3.1416 the
product will be equal to the surface of the whole sphere. If the diameter be 36 inches, what is the surface of a sphere? 3.1415926 X 36 X 36 =
4071.504 square inches, • 144= 14:1862 square or superficial feet. What is the surface of a globe 6 feet in diameter ? 3.1415926 X 6 X6=113.0973 square feet. The height of the segment multiplied by the diameter of a sphere x 3.1416 surface of segment. The diameter of the sphere being 12, and the height of the segment 6. Then 6 X 12 X 3•1416 226.1952 = = surface of spheric segment. How do you find the solidity of a sphere? Multiply the cube of the diameter by •5236 for the solidity, or multiply the surface by one-sixth of the diameter for the solidity. What is the solidity of a sphere 36 inches in diameter ? •523598775 X 36 X 36 = 24429.024 solid inches, or 4071:504 X 6= 24429.024 solid inches. What is the solidity of a globe 6 feet in diameter? •5235987 X 63 113.0973 cubic feet. How do you find the solidity of a segment of a sphere? To 3 times the square of the radius of the base of the segment, add the square of its height; multiply the sum by the height, and the product by ·5236 for the solidity. What is the solidity of a spherical segment of which the height is 2 feet, and the radius of the base 6 feet ? (62 x 3 + 22) X 2 X •5236 = 117.286 solid feet.
How many stones of a rectangular form, each 3 feet by 24 feet, will pave a road 40 yards long and six yards
40 X 6 X 9 2160 broad?
= 288 stones. How 3 X 2.5 7.5 many panes of glass, each 18 inches by 14 will be required for 22 windows, each 5 feet by 3 feet 6 inches? 60 x 42 x 22 = 10 X 22 = 220
The 18 x 14 base of a right-angled triangle is 300 feet, and the sum of the other two sides is 1000 feet; what are their lengths ? 3002 • 1000 = 9, the difference between the hypotenuse and the perpendicular, and 1000 +90
1000 - 90 = 545, the hypotenuse ; also
= 455, the perpendicular. How many 3 inch cubes may be cut
1728 out of a 12 inch cube? 123 - 33
27 What will be the expense of gilding a globe of which the diameter is 6 feet, at 31d. per square inch ? 6 feet = 72 inches and 722 x 3.1416 X 3d. = 57001. 1904d. = 2371. 10s. 1•19d. the expense. How many cubic yards of earth will be dug out in forming a pond, the length being 120 yards, the breadth 60 yards, and the depth 2 yards ? 120 X 2 X 60 = 14400 yards. H