9 to find the number of shingles. Thus, in the last example, the total area in square feet is 40×16×2 1,280 sq. ft., and 1,280 × 9 = 11,520 shingles, the same result as before. = EXAMPLES FOR PRACTICE. 48. Solve the following: 1. How many shingles are required for a roof which measures 45' x 17' on one side and 45′ × 24' on the other side, the exposed portion of the shingles being 4" X 5" ? Ans. 13,284 shingles. 2. (a) How many thousand feet of lumber are contained in a pile having 42 layers of boards 16 feet long, the width of the layers being 11 feet, and the thickness of the boards, 1 inch? (b) What would be its cost at $18.75 per M. Ans. (a) 7.392 M. (b) $138.60. 3. What is the area in square feet of a parallelogram whose base is 581" and altitude is 23g"? Ans. 9.5566+ sq. ft. 4. How many square yards of oilcloth will cover a floor 15' × 13}'? Ans. 221 sq. yd. 5. If Brussels carpet costs 95 cents per yard, what will be the cost of carpeting a room 13' X 18', allowing 1 ft. on each strip for waste in matching? Ans. $36.10. 6. How many sheets of tin 20′′ × 14′′ are required to cover a roof 56' x 30' ? Ans. 864 sheets. 7. At 18 cents per square yard, what will be the cost of plastering the ceiling and walls of a room 23 ft. long, 16 ft. wide, and 12 ft. high, making allowance for 3 doors, 3 ft. 6 in. wide by 7 ft. 6 in. high, 5 windows, 3 ft. 6 in. wide by 5 ft. 4 in. high, and a baseboard 8 in, high? Ans. $21.74. 8. At $2.50 per square yard, what is the cost of paving a street mile long and 60 feet wide? Ans. $44,000. 9. How many double rolls of paper and border are required to cover the walls of the room of example 7, assuming that the border, which is 18 in. wide, extends the height of the baseboard over the paper? Use rule I, Art. 37. Ans. { rolls for border. 9 rolls for walls. 10. How many board feet in a stick of timber 27′ × 9" X 8" ? Ans. 162 ft. 11. How many single rolls of paper would be required to paper the ceiling of the room of example 7, assuming that there is no border, and that the paper overlaps on the walls at least 2 in.? Ans. 11 rolls. THE TRIANGLE. 49. A triangle is a plane figure having three sides. FIG. 18. FIG. 19. 50. An isosceles triangle is one having two of its sides equal, as in Fig. 18. 51. An equilateral triangle is one having all of its sides equal. (Fig. 19.) FIG. 20. FIG. 21. 52. A scalene triangle is one having no two of its sides equal. (Fig. 20.) 53. A right-angled triangle is any triangle having one right angle. The side opposite the right angle is called the hypotenuse. (Fig. 21.) A right-angled triangle may be isosceles or scalene. 54. The altitude of any triangle is a line drawn from B the vertex of the angle FIG. 23. triangle is the sum of the lengths of the three sides. 126D-11 55. If in any parallelogram a straight line, called the diagonal, is drawn, connecting two opposite corners, the parallelogram is divided into two equal triangles, as DAB and DCB, Fig. 24. The area of each triangle, therefore, is equal to onehalf the area of the parallelogram, or to one-half the product of the base and the altitude. Any side of a triangle may be taken as the base. FIG. 24. 56. Rule.-To find the area of a triangle, multiply the base by the altitude and divide the product by 2. EXAMPLE. The base of a triangle is 36 inches long and its altitude is 20 inches. What is the area of the triangle? 57. A circle is a figure bounded by a curved line, called the circumference, every point of which is equally distant from a point within, called the center. (Fig. 25.) The circumference of a circle is also called its periphery. FIG. 25. NOTE. When a surface is bounded by straight lines, the length of the bounding line is called the perimeter; when the bounding line is a curve, the length of the curve is called the periph ery. Thus, we speak of the perimeter of a poly gon, and the periphery of a circle. 58. The diameter of a circle is a 4 straight line passing through the center and terminated at both ends by the circumference. (See AB, Fig. 26.) 59. The radius of a circle is a straight line drawn from the center to the circumference. It is equal in length to one-half the diameter. The plural of radius is radii, and we say that all radii of a circle are equal. (OA, Fig. 27, is a radius.) FIG. 26. FIG. 27. 60. If a circle is divided by a diameter, each half is called a semicircle, and each half-circumference is called a semi-circumference. 61. It has been found that the length of the circumference of any circle divided by the length of the diameter gives a constant number. This number is very nearly 3.1416; it is generally denoted by the Greek letter (pronounced pi). 62. Rule. To find the circumference of a circle, multiply the diameter by 3.1416. Let The above rule may be expressed by the formula, С = π D = 3.1416 D. EXAMPLE.-If a car wheel is 36 inches in diameter, what is its circumference? SOLUTION. C 3.1416 D = = 3.1416 X 36 =113.0976 in. Ans. 63. Rule. To find the diameter of a circle, divide the circumference by 3.1416. Formula: C C D = EXAMPLE. The circumference of a tree is 10 feet 4 inches; what is the diameter? SOLUTION. 10 ft. 4 in 124 in. Using the formula, D: = 64. Rule. To square of the radius diameter by .7854. find the area of a circle, multiply the by 3.1416, or multiply the square of the in which A denotes the area of the circle. EXAMPLE. If the diameter of a circular piston is 14 inches, what is its area? SOLUTION.-The radius is one-half the diameter (Art. 59), or 7 in. EXAMPLES FOR PRACTICE. 65. Solve the following examples: 1. Find (a) the circumference and (b) the area of a circle 34 feet in diameter. Ans. {(6) 907.92 sq. ft. 106.814 ft. Ans. 2,332.834 sq. in. 2. What is the area of a circle 4 feet 6 inches in diameter? rods, of a circular race 3. (a) What must be the diameter, in track 1 mile in length? (b) What is the area of the field enclosed? 4. Find (a) the circumference and (b) the driving wheel, the diameter of which is 5 feet 6 Ans. THE PRISM AND CYLINDER. 66. A solid, or body, has three dimensions: length, breadth, and thickness. The sides that enclose it are called its faces, and the intersections of the sides are called the edges. 67. A prism is a solid whose ends are equal and parallel plane figures, and whose sides are parallelograms. Prisms take their names from the form of their bases. Thus, a triangular prism is one having a triangle for its base. FIG. 28. 68. A parallelopipedon is a prism whose bases (ends) are parallelograms. (Fig. 28.) FIG. 29. 69. A cube is a prism whose faces are equal squares. (Fig. 29.) All the faces of a cube are equal. A cube is also a parallelopipedon. |