AREAS LAND MEASURE 25. In surveying the public lands of the United States and Canada, all linear measurements are made with the surveyors' chain, also known as Gunter's chain, from the name of the inventor. This chain is 66 feet in length and contains 100 links, each 7.92 inches long. In private surveys, the foot is commonly taken as the unit of linear measure, and small land areas are expressed in square feet. Land areas of considerable extent in the countries mentioned are generally expressed in acres. Fractional parts of an acre, which formerly were expressed in roods, square rods or perches, and square links, are now expressed decimally by nearly all surveyors. Thus, 40.35 acres is written instead of 40 acres, 1 rood, and 16 square rods. Tables of linear and square measure are given in Arithmetic, and to those tables the student is referred for detailed information regarding the subject. The following table gives the relative values of the units of area used in land surveying in the countries referred to above. As already stated, the square foot and acre are now the units most commonly employed. *Sometimes called a perch or pole, and designated by the abbreviation P. As will be observed, there are 10 square chains in an acre. In order, therefore, to reduce to acres any number of square chains, it is sufficient to move the decimal point one place toward the left, which is equivalent to dividing by 10. It must also be borne in mind that, since there are 100 links in 1 chain, links are usually expressed decimally as hundredths of a chain. Thus, 6.72 chains is written instead of 6 chains 72 links. EXAMPLE 1.-A rectangular piece of land is 1,060 feet in length by 820 feet in breadth; what is its area: (a) in acres and decimals? (b) in acres, roods, and perches? SOLUTION.- (a) 1,060 x 820 = 19.954 A. Ans. = (b) .954 A. = .954 X 4 = 3.816 R.; .816 R. is 32.64 P. Hence, the area is 19 A. 3 R. 32.64 P. equal to .816 X 40 Ans. EXAMPLE 2.-A rectangular piece of land is 12 chains and 6 links (12.06 chains) in length by 8 chains and 55 links (8.55 chains) in breadth; what is its area: (a) in acres and decimals? (b) in acres, roods, and perches? (b) .311 A. = 103.11 sq. ch.; 103.11 ÷ 10 .311 X 4 = 1.244 R.; .244 R. is equal to .244 X 40 = 9.76 P. Hence, the area is 10 A. 1 R. 9.76 P. Ans. EXAMPLES FOR PRACTICE 1. A rectangular piece of land is 1,190 feet in length by 700 feet in breadth; what is its area: (a) in acres and decimals? (b) in acres, roods, and perches? (a) 19.123 A. Ans. {(6) 19 A. O R. 19.7 P. (b) in acres, roods, and 2. A rectangular piece of land is 525 feet long by 250 feet wide; what is its area: (a) in acres and decimals? perches? (a) 3.013 A. Ans. {) 3 A. 0 R. 2.08 P. 3. A rectangular piece of land is 15 chains and 65 links in length by 8 chains and 16 links in breadth; what is its area: (a) in acres and decimals? (6) in acres, roods, and perches? (a) 12.77 A. Ans. {(6) 12 A. 3 R. 3.2 P. AREAS OF POLYGONS THE TRIANGLE NOTE. In all that follows, the area of any figure under consideration will be designated by S, unless otherwise stated. 26. Given the Base and Altitude.-Any of the sides of a triangle may be taken as the base, the altitude being the length of the perpendicular drawn on the base from the ver tex of the opposite angle. In Fig. 17, b is taken as the base, and the perpendicular BH, denoted by h, B a FIG. 17 27. Given Two Sides and the Included Angle.-Let b, c, and A, Fig. 17, be given. In the right triangle AB H, we have h = c sin A. The substitution of this value of h in the formula in Art. 26 gives Sbc sin A In words, the area of a triangle is equal to one-half the product of any two sides and the sine of their included angle. EXAMPLE.-Two of the sides of a triangular field are 39.47 and 59.23 chains, respectively, and their included angle is 65° 10′ 40′′. To find the contents of the field, in acres. = SOLUTION.-By the formula, S (square chains) 39.47 X 59.23 sin.65° 10′ 40′′′ = 1,060.9 sq. ch.; whence, dividing by 10 (Art. 25), S (acres) = 106.09 A. Ans. 28. Given One Side and Two Angles.-The other angle may be at once found by subtracting the sum of the two given angles from 180°. It may, therefore, be assumed that the three angles are known. Let b, Fig. 17, be the given side. From Art. 22, the value of c is equal to the modulus of the triangle multiplied by sin C, or, Substituting this value in the formula in Art. 27, we obtain 29. The formula in Art. 28 is convenient when logarithmic functions are employed. For the use of natural functions, the following is preferable: In the right triangles A B H and CBH, Fig. 17, we have, AH = h cot A, CH = h cot C whence, adding these two equations, AH+CH = h cot A+ h cot C that is, This formula is useful and should be committed to memory. It may be stated in words thus: The altitude of a triangle is equal to the base divided by the sum of the cotangents of the adjacent angles. By substituting, in the formula in Art. 26, the value of h given in formula 1, we obtain In words, the area of a triangle is equal to the square of any side divided by twice the sum of the cotangents of the angles adjacent to that side. EXAMPLE.-One side of a triangular field is 127.64 chains, and the adjacent angles are 46° 15′ and 60° 41′. To find the area. SOLUTION BY LOGARITHMIC FUNCTIONS.-Here, b 127.64, A = 46° 15', C = 60° 41′, and B 180° 46° 15′- 60° 41′ = 73° 4′. Formula of Art. 28, = 127.64 sin 46° 15′ sin 60° 41′ 5,363.4 sq. ch. SOLUTION BY NATURAL FUNCTIONS.-By formula 2, S = = 2(cot 46° 15' + cot 60° 41') 2(.95729 + .56156) = 3.0377 5,363.4 sq. ch. = 536.34 A. Ans. NOTE. Even if natural functions are used, the division is advantageously performed by means of logarithms. EXAMPLES FOR PRACTICE 1. Two sides of a triangular field are 3,760 and 2,757 feet, respectively, and their included angle is 54° 13′ 13′′. What is the area of the field, in acres? Ans. S 96.534 A. 2. One side of a triangle is 96.34 chains; the opposite angle is 49° 10′, and one of the adjacent angles, 69° 45′ 30′′. the triangle, in acres? What is the area of Ans. S 503.69 A. 3. One side of a triangle is 8.93 inches, and the adjacent angles are 34° 16' and 17° 37′ 18′′. What is the area of the triangle? Ans. S 8.638 sq. in. 4. Two sides of a triangle are 17 and 25 feet, respectively, and the included angle is 76° 13'. What is the area of the triangle? 30. Given the Three Sides.—Let a, b, and c, Fig. 17, be given, and denote (a+b+c) by s. The area S of the triangle is given by the following formula, which is derived in Appendix VI: EXAMPLE.-The sides of a triangular tract are 1,634.6 (= a, say), 978.28 (= b, say), and 2,176.4 (= c, say) feet, respectively; to find the area, in acres. SOLUTION.-The work may be conveniently arranged as shown below. The numbers in marks of parenthesis indicate the order in which the several quantities are set down. In (6), s is placed above a, b, c in order to facilitate the subtractions. The differences s-a, s-b, c are written, as the subtractions are performed, horizontally opposite a, b, and c, respectively. 5.87509; S = 750,050 sq. ft. 17.22 A. Ans. |