CHAPTER XIV. RULES OF MENSURATION FOR THE CALCULATION OF AREAS AND VOLUMES. 1. AREA of rectangle, as ABCD (Fig. 148).—The area is given by AB X BC. The length and breadth must be of the same denomination, i.e. if the length is in feet the breadth must be in feet, the area then being square feet. 2. Area of triangle, as ABC (Fig. 149).—We draw CD perpendicular to the base AB, meeting it, or it produced, in D. Then the area AB x CD, or one-half the base into the height. 3. Area of trapezoid.-This is a four-sided figure, in which = two sides only are parallel, as ABCD (Fig. 150). Calling the parallel sides a and b respectively, and h the distance between them, the area = (a + b) h, or one-half the sum of the parallel sides multiplied by the distance between them. 4. Circle.-(a) Length of circumference is π times the diameter, where 3.1416, or 22 nearly. 5. Curvilinear figure, as ABCD (Fig. 151).—The area of figures of this character are continually required in ship calculations. (a) Trapezoidal rule. This rule has found considerable favour, especially in France and the United States, on account of its great simplicity. We divide the base into a number of equal spaces and erect ordinates to meet the curve as EF, GH, etc. Calling the common interval h and the length of ordinates y1, Y2, · have, regarding the figures ADFE, EFHG, etc., as trapezoids Adding all together- (y2 + ya) h, and so on to (Y6 + y7) h. ... Y7, we h (y1 +22 + 2Y3 + 2y + 21⁄45 + 21⁄46 + Y7) = h ( Y1 + Y 7 + Y 2 + Y s + Y + Y5 + Ye), 2 i.e. the first and last ordinates are added together and divided by two, then all the remaining ordinates are added and the total sum is multiplied by the common interval. The following example will illustrate the use of the rule and how near the result obtained is to the real area required. The curve, whose ordinates 2 ft. apart are 0, 2.2, 4·0, 5·4, 64, 7·0, and 7.2 ft. respectively, is a portion of a common parabola, and the exact area enclosed is 57'6 square ft. Find the area by using the trapezoidal rule. 0 +7.2 = 57.2 square ft. There is thus an error of nearly 1 per cent. The error involved in using this rule is lessened by spacing the ordinates closely, but it is not used in Admiralty calculations on account of the approximate nature of the results obtained. The rule that is employed is (b) Simpson's first rule.-Take a figure as ABCD (Fig. 152), and divide the base into two equal parts in the point E. Then, assuming the curve is a common parabola, the area ABCD is h given by (1 + 42 + Yз). To apply this rule to a longer figure, 3 as ABCD (Fig. 151), we divide the base into an even number of intervals, so that the above rule may be applied to each portion containing a a pair of intervals, The multipliers thus are 1, 4, 2, 4, 2, 4, 1, and there must be an even number of intervals or an odd number of ordinates for the rule to be applicable. In working by this rule it is advisable to use a table as follows, which is the calculation for the area of the figure considered above, the exact area of which is known to be 57.6 square ft. The area given by using this rule is seen to be the exact area. This is necessarily so because the curve is a common parabola, and this is the curve on which the rule is based. M 13. EXAMPLE. The following ordinates, 1-3 ft. apart, give a curve which is an arc of a circle of 6 ft. radius, viz. 0, 1.56, 2·41, 2·86, 3.00, 2.86, 2:41, 1.56, 0. The exact area of the circular segment thus obtained is 22.1 square ft. If the area be calculated by the two methods considered above we have (a) Area by trapezoidal rule, 21-64 square ft. (b) Area by Simpson's rule, 22.1 It is thus seen that the latter gives a correct result, even though the curve is not a parabola. Also the trapezoidal rule is in error to the extent of over 2 per cent. Volumes.-To find the volume of a solid bounded by a curved surface. The volumes of such bodies as this are continually required CURVE OF SECTIONAL AREAS. in ship calculations, the most important case being the underwater volume of a ship. The method adopted is to divide the volume by a series of equidistant planes; we then find the area of each of the figures traced out on these planes by the surface, as in Fig. 154, and treat these areas as the ordinates of a curve having the same length as the body. The area enclosed by this "curve of areas" will give the volume required. In finding the underwater volume of a ship we may divide the volume in two ways, viz. 1. By means of equidistant planes in transverse sections. The shape of these sections is given in the "body plan" in Fig. 155, and in perspective in Fig. 154. |