Section I. Areas of figures bounded by right lines, II. The Quadrature of the Circle and its parts, Promiscuous examples of Areas, III. Solids bounded by plane surfaces, IV. The Cylinder, Cone, and Sphere, SECTION I. AREAS OF FIGURES BOUNDED BY RIGHT LINES. ART. 1. THE following definitions, which are nearly the same as in Euclid, are inserted here for the convenience of reference. I. Four-sided figures have different names, according to the relative position and length of the sides. A parallelogram has its opposite sides equal and parallel; as ABCD. (Fig. 2.) A rectangle, or right parallelogram, has its opposite sides equal, and all its angles right angles; as AC. (Fig. 1.) A square has all its sides equal, and all its angles right angles; as ABGH. (Fig. 3.) A rhombus has all its sides equal, and its angles oblique; as ABCD. (Fig. 3.) A rhomboid has its opposite sides equal, and its angles oblique; as ABCD. (Fig. 2.) A trapezoid has only two of its sides parallel; as ABCD. (Fig. 4.) Any other four sided figure is called a trapezium. II. A figure which has more than four sides is called a polygon. A regular polygon has all its sides equal, and all its angles equal. III. The height of a triangle is the length of a perpendicular, drawn from one of the angles to the opposite side; as CP. (Fig. 5.) The height of a four sided figure is the perpendicular distance between two of its parallel sides; as CP. (Fig. 4.) IV. The area or superficial contents of a figure is the space contained within the line or lines by which the figure is bounded.. 2. In calculating areas, some particular portion of surface is fixed upon, as the measuring unit, with which the given figure is to be compared. This is commonly a square; as a square inch, a square foot, a square rod, &c. For this reason, determining the quantity of surface in a figure is called squaring it, or finding its quadrature; that is, finding a square or number of squares to which it is equal. 3. The superficial unit has generally the same name, as the linear unit which forms the side of the square. The side of a square inch is a linear inch; of a square foot, a linear foot; of a square rod, a linear rod, &c. There are some superficial measures, however, which have no corresponding denominations of length. The acre, for instance, is not a square which has a line of the same name for its side. The following tables contain the linear measures in common use, with their corresponding square measures. An acre contains 160 square rods, or 10 square chains. By reducing the denominations of square measure, it will be seen that 1 sq. mile=640 acres=102400 rods=27878400 feet=4014489600 inches. The fundamental problem in the mensuration of superficies is the very simple one of determining the area of a right parallelogram. The contents of other figures, particularly those which are rectilinear, may be obtained by finding parallelograms which are equal to them, according to the principles laid down in Euclid. PROBLEM 1. To find the area of a PARALLELOGRAM, square, rhombus, or rhomboid. 4. MULTIPLY THE LENGTH BY THE PERPENDICULAR HEIGHT OR BREADTH. It is evident that the number of square inches in the parallelogram AC (Fig. 1.) is equal to the number of linear inches in 'the length AB, repeated as many times as there are inches in the breadth BC. For a more particular illustration of this see Alg. 511-514. The oblique parallelogram or rhomboid ABCD, (Fig. 2.) is equal to the right parallelogram GHCD. (Euc. 36. 1.) The area, therefore, is equal to the length AB multiplied into the perpendicular height HC. And the rhombus ABCD, (Fig. 3.) is equal to the parallelogram ABGH. As the sides of a square are all equal, its area is found, by multiplying one of the sides into itself. Ex. 1. How many square feet are there in a floor 23 feet long, and 18 feet broad? Ans. 23×18-423. 2. What are the contents of a piece of ground which is 66 feet square? Ans. 4356 sq. feet 16 sq. rods. = 3. How many square feet are there in the four sides of a room which is 22 feet long, 17 feet broad, and 11 feet high? Ans. 858. ART. 5. If the sides and angles of a parallelogram are given, the perpendicular height may be easily found by trigonometry. Thus, CH (Fig. 2.) is the perpendicular of a right angled triangle, of which BC is the hypothenuse. Then, (Trig. 134.) R BC sin B : CH. : The area is obtained by multiplying CH thus found, into the length AB. Or, to reduce the two operations to one, As radius, To the sine of any angle of a parallelogram ; For the area=ABXCH, (Fig. 2.) But CH= BCxsin B R Therefore, ABXBCxsin B The area: Or, R: sin B:: ABXBC; the area. Ꭱ Ex. If the side AB be 58 rods, BC 42 rods, and the angle B 63, what is the area of the parallelogram? 2. If the side of a rhombus is 67 feet, and one of the angles 730, what is the area? Ans. 4292.7 feet. 6. When the dimensions are given in feet and inches, the multiplication may be conveniently performed by the arithmetical rule of Duodecimals; in which each inferior denomination is one twelfth of the next higher. Considering a foot as the measuring unit, a prime is the twelfth part of a foot; a second, the twelfth part of a prime, &c. It is to be observed, that, in measures of length, inches are primes; but in superficial measure they are seconds. In both, a prime is of a foot. But of a square foot is a parallelogram, a foot long and an inch broad. The twelfth part of this is a square inch, which is of a square foot. Ex. 1. What is the surface of a board 9 feet 5 inches, by 2 feet 7 inches. 2. How many feet of glass are there in a window 4 feet 11 inches high, and 3 feet 5 inches broad? Ans. 16 F. 9' 7", or 16 feet 115 inches. 7. If the area and one side of a parallelogram be given, the other side may be found by dividing the area by the given side. And if the area of a square be given, the side may be found by extracting the square root of the area. This is merely reversing the rule in art. 4. See Alg. 520, 521. Ex. 1. What is the breadth of a piece of cloth which is 36 yds. long, and which contains 63 square yds. Ans. 13 yds. |