= and BM= BL, CK+CM will be equal to the sum of the sides AB, AC, and BC; therefore CK or CM: (AB+ AC+ BC) = S, if S stand for the sum of the three sides. But CE + AE+ BG =S; therefore CK = CM = CABG, and AK = AL = BG; whence AG = AE = BL = BM, and EK: AB. Now, since CK= CMS, we have AKS — AC, EC = and AE BMS — BC. = = S-AB, Because the triangles CDE and CKH, as also ADE and HKA, are similar, Now, ABC= ACD+ BCD + ABD = 1⁄2 AC. ED + 1⁄2 BC . ED + 1⁄2 AB. ED =S. ED CK. ED. COR. From the above demonstration, it is apparent that the area of a triangle is equal to the rectangle of the half-sum of the sides and the radius of the inscribed circle. For another demonstration of this rule, see Appendix. EXAMPLES. Ex. 1. Required the area of a triangle, the three sides being 672, 875, and 763 links respectively. NOTE.—In cases of this kind the operation will be much facilitated by using logarithms. Ex. 2. Required the area of a triangular tract, the sides of which are 17.25 chains, 16.43 chains, and 14.65 chains respectively. Ans. 11 A., 0 R., 14.4 P. Ex. 3. Given the three sides, 19.58 chains, 16.92 chains, and 12.76 chains, of a triangular field: required the area. Ans. 10 A., 2 R., 27 P. 252. Trapezoids. Measure the parallel sides and the perpendicular distance between them. If a plat is desired, a diagonal, or the distance AE, (Fig. 105,) may be measured. Multiply the sum of the parallel sides by half the perpendicular: the product is the area. Fig. 105. D DEMONSTRATION. - ABCD = ABD + BCD = AB. DE DC . DE = (AB+ DC). DE. EXAMPLES. Ex. 1. Given AB = 7.75 chains, DC = 5.47 chains, and DE = 4.43 chains, to calculate the content and plat the map, AC being 7.00 chains. Ans. Area, 2 A., 3 R., 28.5 P. Ex. 2. Given the parallel sides of a trapezoid, 16.25 chains and 14.23 chains, respectively: the perpendicular from the end of the shorter side being 12.76 chains, and the distance from the foot of said perpendicular to the adjacent end of the longer side 1.37 chains. Required the area and plat. Ans. 19 A., 1 R., 31.4 P. 253. Trapeziums. First Method.-Measure a diagonal, and the perpendiculars thereon, from the opposite angle. The area of a trapezium is equal to the rectangle of the diagonal and half the sum of the perpendiculars from the opposite angles. This is evident from the triangles of which the trapezium is composed. EXAMPLES. Ex. 1. To plat and calculate the area of a trapezium, the diagonal being 15.63 chains, and the perpendiculars thereto from the opposite angles being 8.97 and 6.43 chains, and meeting the diagonal at the distances of 4.65 and 13.23 chains. Ans. Area, 12 A., 0 R., 5.6 P. Ex. 2. Given (Fig. 106) AC = 19.68 chains, AE = 7.84 chains, AF = 16.23 chains, ED 10.42 chains, and FB = 8.73 chains, to plat the figure and find the area. = Ans. 18 A., 3 R., 14.98 P. Ex. 3. Required the area of a trape Fig. 106. B E A C zium, the diagonal being 17.63 chains, and the perpendiculars 6.47 and 12.51 chains respectively. Ans. 16 A., 2 R., 36.94 P. 254. Second Method.-Measure one side, and the perpen diculars thereon from the extremities of the opposite side, with the distances of the feet of these perpendiculars from one end of the base. 10 Lay off the distances AE, AF, and AB; then erect the perpendiculars ED and FC, and draw AD, DC, and CB. The trapezium is divided into two triangles and the trapezoid, the areas of which may be found by the preceding rules. Thus, 2 AED = AE.ED = 23.8383 2 EFCD = EF. (ED+FC) = 141.3120 2 CFB = whence ABCD CF. FB = 98.2655 of 263.4158 = 131.7079 chains 13 A., 0 R., 27.3 P. = If either of the angles A or B were obtuse, the perpendicular would fall outside the base, and the area of the corresponding triangle should be subtracted. Ex. 2. Plat and calculate the area of a trapezium from the following field-notes: Ex. 3. Calculate the area from the following field Fields of more than four sides, bounded by 255. First Method.-Divide the tract into triangles and trapeziums, and calculate the areas by some of the preceding rules. In applying this method, as many of the measurements as practicable should be made on the ground; the field then being platted with care, the other distances may be measured on the map. When it is intended to depend on the map for the distances, every part of the plat should be laid down with scrupulous accuracy, on a scale of not less than three chains to the inch. Ex. 1. To draw the map and calculate from the following field-notes the area of the pentagonal field ABCDE: Ex. 2. Map the plat, and calculate the content from the following fieldnotes: |