525. To find the area of a trapezoid. 1. Find the area of a trapezoid whose parallel sides are 23 ft. and 11 ft., and the altitude 9 ft. OPERATION. 23 ft.+ 11 ft. ÷ 2 = 17 ft.; 17 ft. x 9 = 153 sq. ft., area. RULE. Multiply one half the sum of the parallel sides by the altitude. 2. What is the area of a trapezoid whose parallel sides are 178 ft. and 146 ft., and the altitude 69 ft. Ans. 11178 sq. ft. 3. How many square feet are there in a board 16 ft. long, 18 in. wide at one end and 25 in. wide at the other end? 4. One side of a quadrilateral field measures 38 rd.; the side opposite and parallel to it measures 26 rd., and the distance between the two sides is 10 rd. Find the area. 526. To find the area of a trapezium. 1. Find the area of a trapezium whose diagonal is 42 ft. and perpendiculars to this diagonal, as in the diagram, are 16 ft. and 18 ft. OPERATION. 42 ft. 18 ft. (18 ft.+ 16 ft. ÷ 2) × 42 = 714 sq. ft., area. Ans. 2 A. 16 ft. RULE. Multiply the diagonal by half the sum of the perpendiculars drawn to it from the vertices of the opposite angles. 2. Find the area of a trapezium whose diagonal is 35 ft. 6 in., and the perpendiculars to this diagonal 9 ft. and 12 ft. 6 in. 3. How many acres are there in a quadrilateral field whose diagonal is 80 rd. and the perpendiculars to this diagonal 20.453 and 50.832 rd. ? 1. To find the area of any regular polygon, multiply its perimeter, or the sum of its sides, by the perpendicular falling from its center to one of its sides. 2. To find the area of an irregular polygon, divide the figure into triangles and trapeziums, and find the area of each separately. The sum of these areas will be the area of the whole polygon. CIRCLES. 527. A Circle is a plane figure bounded by a curved line, called the circumference, every point of which is equally distant from a point within called the center. 528. The Diameter of a circle is a line passing through its center, and terminated at both ends by the circumference. 529. The Radius of a circle is a line extending from its center to any point in the circumference. It is one half the diameter. EXAMPLES. 530. When either the diameter or the circumference of a circle is given, to find the other dimension of it. 1. Find the circumference of a circle whose diameter is 20 in. OPERATION.-20 in. x 3.141662.832 in. 5 ft. 2.832 in., cir cumference. = 2. Find the diameter of a circle whose circumference is 62.832 ft. OPERATION. -62.832 ft. ÷ 3.1416 20 ft., diameter. = RULE.-I. Multiply the diameter by 3.1416; the prod uct is the circumference. II. Divide the circumference by 3.1416; the quotient is the diameter. 3. What is the circumference of a wheel 5 ft. 6 in. in diameter ? 4. Find the diameter of a wheel whose circumference is 50 ft. 5. What is the diameter of a tree whose girt is 18 ft. 6 in.? 6. Find the length of a tire that will band a wheel 7 ft. 9 in. in diameter. Ans. 24 ft. 4+ in. 7. The diameter of a cylinder is 8 ft. 6 in. Find its girt. 8. What is the radius of a circle whose circumference is 31.416 ft.? 531. To find the area of a circle, when both its diameter and circumference are given, or when either is given. 1. Find the area of a circle whose diameter is 10 ft. and circumference 31.416 ft. OPERATION.-31.416 ft. x 10 ÷ 4 = 78.54 sq. ft., area. 2. Find the area of a circle whose diameter is 10 ft. OPERATION. (10 ft.)2 x .7854 78.54 sq. ft., area. 3. Find the area of a circle whose circumference is 31.416 ft. OPERATION.-31.416 ft.÷3.1416=10 ft., diam.; (10 ft.)2 ×.7854 78.54 sq. ft., area. RULE. -I. Multiply the square of its diameter by .7854. II. Multiply of its diameter by the circumference. 4. What is the area of a circular pond whose circumference is 200 chains? Ans. 318.3 A. 5. The distance around a circular park is 1 miles. How many acres does it contain ? Ans. 114.59 A. 6. Find the area of the largest circle that can be drawn by using as a radius a string 20 in. long. 532. To find the diameter or circumference of a circle, when the area is given. 1. What is the diameter of a circle whose area is 1319.472 ? OPERATION. - 1319.472÷.7854=1680; √1680=40.987+, diam. 2. What is the circumference of a circle whose area is 19.635 ? RULE. -I. Divide the area by .7854 and extract the square root of the quotient; the result is the diameter. II. Divide the area by 3.1416 and extract the square root of the quotient; the result is the radius. The circumference is obtained by 530. Or, III. Divide the area by .07958, and extract the square root of the quotient. 3. The area of a circular lot is 38.4846 square rods. What is its diameter ? 4. The area of a circle is 286.488 square feet. What are its diameter and the circumference ? 5. The area of a circular lot is 1 acre. What is its diameter ? } = area. Multiplied by {7854, the quotient 1.2732, the product Divided 7. The square of the radius of any circle x 3.1416 SOLIDS. 533. A Solid or Body has three dimensions, length, breadth, and thickness. The planes which bound it are called its faces, and their intersections, its edges. PRISMS AND CYLINDERS. 534. A Prism is a solid whose ends are equal and parallel polygons, and its sides parallelograms. Prisms take their names from the forms of their bases, as triangular, quadrangular, pentagonal, etc. 535. The Altitude of a prism is the perpendicular distance between its bases. 536. A Parallelopipedon is a prism bounded by six parallelograms, the opposite ones being parallel and equal. 537. A Cube is a parallelopipedon whose faces are all equal squares. Parallelopipedon. Cube. 538. A Cylinder is a body bounded by a uniformly curved surface, its ends being equal and parallel circles. 1. A cylinder is conceived to be generated by the revolution of a rectangle about one of its sides as an axis. 2. The line joining the centers of the bases, or ends, of the cylinder is its altitude, or axis. |