462. Quadrilaterals are of three kinds, as follows: Parallelogram. Trapezoid. Trapezium. A parallelogram has its opposite sides parallel; a trapezoid has only two sides parallel; and a trapezium has no two sides parallel. The diagonal of a quadrilateral, as shown in the figures, is a line joining any two angles not adjacent. 463. Parallelograms are of four kinds, as follows: Square. Rectangle. A square has all its sides equal, and all its angles right angles; a rectangle has its opposite sides equal, and all its angles right angles; a rhomboid has its opposite sides equal, and its angles acute and obtuse; a rhombus has all its sides equal, and its angles acute and obtuse. An acute angle is less than a right angle; an obtuse angle is greater than a right angle. The altitude of a parallelogram or a trapezoid is the perpendicular distance between its parallel sides. Rhomboid. Rhombus. THE CIRCLE. 464. 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. The diameter of a circle is a line passing through its center, and terminated at both ends by the circum. ference. The radius of a circle is a line extending from its center to any point in the circumference. It is one half the diameter. PROBLEMS. 465. The base and altitude of a triangle given, to find the area. 1. Find the area of a triangle whose base is 26 ft. and altitude 14.5 feet. SOLUTION. 14.5 × 26 1881; hence, the area is 1881 sq. ft. 2 (base x altitude) = area of triangle. = FORMULA: Find the area of a triangle, 2. Whose base is 12 ft. 6 in. and altitude 6 ft. 9 in. 3. Whose base is 25.01 chains and altitude 18.14 chains. 4. What is the cost of a triangular piece of land whose base is 15.48 ch. and altitude 9.67 ch., at $60 an acre? 5. At 8.40 a square yard, find the cost of paving a triangu lar court, its base being 105 feet, and its altitude 21 yards? 189 × 2 466. The area and one dimension given, to find the other dimension. 1. Find the base of a triangle whose area is 189 sq. ft., altitude 14 ft. SOLUTION. =27; hence, the base is 27 ft. 2. Find the altitude of a triangle whose area is 20 sq. ft., base 9 yards. RULE.-Divide twice the area by the given dimension. Find the other dimension of the triangle, 3. When the area is 65 sq. in. and the altitude 10 in. 4. When the base is 42 rods and the area 588 sq. rods. 5. When the area is 6 acres and the altitude 17 yards. 6. When the base is 12.25 chains and the area 5 A. 33 P. 467. The three sides of a triangle given, to find its area. 1. Find the area of a triangle whose sides are 30, 40, and 50 ft. SOLUTION. (30 + 40 + 50) ÷ 2 = 60; 603030; 60 - 40 = 20; 60 - 50 10. √60 × 30 × 20 × 10 = 600 ft., area. 2. Find the area of a triangle whose base is 20 ft. and each of the other sides 15 ft. RULE. From half the sum of the three sides subtract each side separately; multiply the half-sum and the three remainders together; the square root of the product is the area. 3. Find the area of a triangle whose sides are 25, 36, and 49 in. 4. How many acres in a field in the form of an equilateral triangle whose sides measure 70 rods? 468. To find the area of a parallelogram. 1. Find the area of a parallelogram 16.25 ft. by 7.5 ft. wide. SOLUTION. 16.25 × 7.5 = 121.875; hence, 121.875 sq. ft. is the area. FORMULA: Base x altitude area of parallelogram. 2. The base of a rhombus is 10 ft. 6 in., and its altitude 8 ft. What is its area? 3. How many acres in a piece of land in the form of a rhomboid, the base being 8.75 ch. and altitude 6 ch.? 4. A man bought a farm 198 rods long and 150 rods wide, and agreed to give $32 an acre. What did the farm cost? 5. A certain rectangular piece of land measures 1000 links by 100. How many acres does it contain?· 6. How many square feet in a board 16 ft. long, 18 inches wide at one end and 25 inches wide at the other end? SOLUTION. 469. To find the area of a trapezoid. 1. Find the area of a trapezoid whose parallel sides are 23 and 11 ft., the altitude 9 ft. FORMULA: (sum of the bases altitude)=area of trapezoid. 2. Required the area of a trapezoid whose parallel sides are 178 and 146 feet, and the altitude 69 feet. 3. One side of a quadrilateral field measures 38 rods; the side opposite and parallel to it measures 26 rods, and the distance between the two sides is 10 rods. Find the area. 470. 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. SOLUTION. 18+16 × 42 = 42 ft. /18 ft. 714; hence, 714 sq. ft. is the area. 16 ft FORMULA: (sum perpendic. × drag.) = area of trapezium. 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. 3. How many acres in a quadrilateral field whose diagonal is 80 rd. and the perpendiculars to this diagonal 20.453 and 50.832 rd.? To find the area of any regular polygon, multiply its perimeter, or the sum of its sides, by the perpendicular falling from its centre to one of its sides. 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. 471. The diameter or circumference of a circle given, to find the other dimension. 1. Find the circum. of a circle whose diameter is 20 in. SOLUTION.-20 in. x 3.1416 62.832 in., the circumference. 2. Find the diameter of a circle whose circumference is 62.832 ft. SOLUTION.-62.832 ft. 3.1416 20 ft., the diameter. 1. Diameter x 3.1416 FORMULAS circumference. 2. Circumference ÷ 3.1416 = diameter. 3. Find the diameter of a wheel whose circum. is 50 ft. 4. What is the diameter of a tree whose girt is 18 ft. 6 in.? 5. What is the radius of a circle whose circum. is 31.416 ft.? 6. Find the circumference of the greatest circle that can be drawn with a string 14 inches long, used as a radius. 472. 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 feet. = SOLUTION.-31.416 ft. x 10÷4 78.54 sq. ft., area. 2. Find the area of a circle whose diameter is 10 ft. SOLUTION.-10 ft. x .7854 = 78.54 sq. ft., area. 3. Find the area of a circle whose circum. is 31.416 ft. SOLUTION.-31.416 ft. ÷ 3.1416 = 10 ft., diam.; (10 ft.)2 × .7854 78.54 sq. ft., area. 1. == (diameter x circumference) = area. { 2. Square of diameter ×.7854 = area. FORMULAS: 4. Find the area of a circular pond, its circum. being 200 ch.? 5. The distance around a circular park is 1 miles. How many acres does it contain ? 6. How much land in a circular garden, that requires 84 rd. of fencing to inclose it? |