QUADRILATERALS. 412. A Quadrilateral is a polygon having four sides, and therefore four angles. A Parallelogram is a quadrilateral having its opposite sides parallel. A RECTANGLE is a right-angled parallelogram. A SQUARE is a rectangle having equal sides. A PARALLELOGRAM. A SQUARE. A RHOMBOID is a parallelogram having no right angles. A RHOMBOID. A RHOMBUS. 414. A Trapezoid is a quadrilateral having only two of its sides parallel. 415. A Trapezium is a quadrilateral having no two of its sides parallel. A TRAPEZOID. A TRAPEZIUM. AREAS OF TRIANGLES AND QUADRILAT ERALS. 416. By Geometry may be proved, in relation to areas, the following PRINCIPLES. 1. The area of a PARALLELOGRAM is equal to the product of the base by the altitude. What is a Quadrilateral? A Parallelogram? A Rectangle? A Rhomboid? A Rhombus? A Trapezoid? A Trapezium? To what is the area of a parallelogram equal? This has been shown to be the case with a rectangle (Art. 210), and that it applies equally to a rhomboid or rhombus, appears from the diagram, in which the rhomboid A B CD is equivalent to the rectangle ABE F, of the same base and altitude. F DE C A B 2. The area of a TRAPEZOID is equal to the product of half the sum of the parallel sides by the altitude. H A D For, any trapezoid A B C D is equivalent CK to a parallelogram A L K D of the same altitude, and whose base AL is equal to H I, which is half of A B+ CD. L I 3. The area of a TRIANGLE is equal to the product of half the base by the Baltitude, or of half the altitude by the base. For, any triangle A B C is equivalent to one half of the parallelogram B C E A, of the same base and altitude. 4. The area of a TRAPEZIUM, or of any polygon, is equal to the sum of the areas of B the triangles into which it may be resolved. Thus, the trapezium W X Y Z is equal to the triangle W X Z plus the triangle X Y Z, made by the diagonal X Z. E D 5. The area of a REGULAR POLYGON is equal to the product of the perimeter by half the perpendicular drawn from the center to any one of the sides. For, any regular polygon, ABCDEF, F may be resolved into as many equal triangles as it has sides, by drawing from the center, O, the lines O A, O B, O C, etc. C N To what is the area of a trapezoid equal? Of a triangle? Of a trapezium ? Of a regular polygon? Exercises. 1. What is the area of a board 18.8 feet long and 2.7 feet wide? Ans. 50.76 sq. ft. 2. What is the area of a board 28 feet long and 15 inches broad? Ans. 35 sq. ft. 3. If the base of a gable of a house be 40 feet long and its perpendicular hight 20 feet, how many square feet of boards will be required to cover it? Ans. 400 sq. ft. 4. How many acres in a triangular lot, one side measuring 32 rods, and the shortest distance from this side to the opposite angle being 14 rods? Ans. 1 A. 64 sq. rd. 5. If the parallel sides of a lot be 75 and 33 yards, and its breadth 20 yards, what is the area in square rods? Ans. 35.7+ sq. rd. 6. How many hectares in a rectangular meadow 640 meters long and 240 meters wide? Ans. 15 hectares and 36 ares. 7. One of the diagonals of a field in the form of a trapezium is 160 rods long, and the perpendiculars from the opposite angles to that diagonal are 70 and 50 rods; what is the area? Ans. 60 acres. 417. When the three sides of a triangle are given, we may, to find the area, Take half the sum of the three sides, subtract therefrom each side separately, multiply together the four results, and extract the square root of the product. 8. The sides of a triangle are 13, 84, and 85 rods, respectively; what is its area? Ans. 3 A. 66 sq. rd. 9. The sides of a certain field in the form of a trapezium measure 30, 35, 40, and 25 rods, respectively, and the diagonal which forms a triangle with the first two sides, 45 rods; what is the area? Ans. 6 A. 61.8 sq. rd. 10. What is the area of a regular hexagon, whose sides When the three sides of a triangle are given, how may the area be found? are each 14.6 feet, and the perpendicular from the center to a side 12.64 feet? Ans. 553.63+ sq. ft. CIRCLES. 418. A Circle is a plane figure bounded by a curved line, all the points of which are equally distant from a point within, called the center. The CIRCUMFERENCE is the bound ing line; as the line A E B D. The DIAMETER is any straight line drawn through the center and terminating in the circumference; as the line A B. The RADIUS is any straight line drawn from the center to the circumference; as the lines CA, CB, or CD. D A D Α B C E 419. A Square is said to be inscribed in a circle when the vertices of its angles are in the circumference. Thus, The square ABCD is inscribed in a circle. 420. By Geometry there may be proved the following PRINCIPLES. 1. The CIRCUMFERENCE of every circle is nearly 3.1416 times its diameter. Hence, 2. The CIRCUMFERENCE is equal to the product of the diameter by 3.1416; and 3. The DIAMETER is equal to the quotient of the circumfer ence divided by 3.1416. What is a Circle? The Circumference? The Diameter ? The Radius ? How many times the diameter is the circumference? To what is the circumference equal? The diameter ? 4. The AREA is equal to the product of the circumference by one half of the radius, or by one fourth of the diameter. Hence, 5. The AREA is equal to the product of the square of the diameter by .7854; and 6. The DIAMETER is equal to the square root of the quotient of the area divided by .7854. 7. The SIDE of every square inscribed in a circle is nearly .7071 times the diameter, or .2251 times the circumference; also, 8. The SIDE of every square inscribed in a circle is equal to the square root of half the square of the diameter. 9. The SIDE of a square equal in area to a given circle is equal to the product of the diameter by .8862. Exercises. 1. What is the circumference of a circle whose diameter is 20 feet? Ans. 62.83+ feet. 2. What is the diameter of a circle whose circumference is 142 yards? Ans. 45.19 yards. 3. What is the area of a circle whose diameter is 100 yards? Ans. 7854 sq. yd. 4. What must be the side of a square stick of timber that can be hewn from a round stick 24 inches in diameter? Ans. 16.97 inches. 5. A wheel is 5 feet in diameter; what is the length of its tire? 6. The area of a circle is 5 acres 146 square rods; what is the diameter? Ans. 34.7 rods. 7. What is the surface in ares of a circular fish-pond which is 50 meters in diameter ? Ans. 19 ares 63.5 centares. To what is the area equal? The side of every inscribed square The side of a square equal in area to a given circle? |