Computation of the Areas of Lots Adjoining the Boundaries of Townships. (Paraphrased from the Manual of Surveying Instructions, 1894)

In regular townships, the tracts of land in each section adjoining the north and west boundaries of such townships, in excess of the regularly subdivided 480 acres (except in section 6), will, in general, be in the form of trapezoids, 80.00 chains in length by about 20 chains in width.

On the plats of such townships, each of said tracts will be divided into four lots, by drawing broken lines at intervals of 20.00 chains, parallel to the ends of the tracts, which will be regarded as parallel to each other.

With the exception of section 6, the south boundaries of sections of the north tier, when within prescribed limits, will be called 80.00 chains.

The areas of the lots in any one tract (except in section 6) may be determined as follows:

Divide the difference between the widths of the ends of the tract by 4; if 3 remains, increase the hundredth figure of the quotient by a unit; in all other cases disregard the fraction; call the quotient thus obtained, “d”; then, taking the end widths of the tract in chains and decimals of a chain, the areas of the lots, in acres, will be:

• Of the smallest lot: twice the width of the lesser end, plus “d”;
• Of the largest lot: twice the width of the greater end, minus “d”;
• Of the smaller middle lot: sum of the widths of the ends, minus “d”;
• Of the larger middle lot: sum of the widths of the ends, plus “d”.

A check on the computation may be had by multiplying the sum of the widths of the ends of the tract by 4: the product should agree exactly with the total area of the four lots.

The proper application of the above rules will always give areas correct to the nearest hundredth of an acre; and, as the use of fractions is entirely avoided, the method is recommended for its simplicity and accuracy.

The 1/4 difference of latitudinal boundaries is 0.0375 chains; consequently, “d”: is 0.04 chains; then:

18.35 X 2+ .04 = 36.74 acres, the area of lot 1;
18.50 X 2 – .04 = 36.96 acres, the area of lot 4;
18.50 + 18.35 – .04 = 36.81 acres, the area of lot 2;
18.50 + 18.35 + .04 = 36.89 acres, the area of lot 3;
Check: (18.35 + 18.50) X 4 = 147.40 acres, the area of the four lots.

The areas of the lots in section 6 may be determined as follows:

The areas of lots 5, 6, and 7 may be obtained by the foregoing rules in all cases, except when the township closes on a base line or standard parallel; also, the area of lot 4, provided both meridional boundaries are 80.00 chains in length; when the last condition obtains, the areas of lots 1, 2, 3, and 4 will be computed as follows:

Refer to the adjacent diagrams and determine +ha difference “a”- between the east boundaries of lots 1 and 4 by the following proportion:

N. bdy. sec. 6.: diff. of meridional bdrs. sec. 6.: 60 chs.:q; then will E. bdy. of lot 4 = E, bdy. lot 1 + q; in which, “q” will be added when the east boundary of sec. 6 is less than 80.00 chains; but subtracted when said easl boundary is greater than 80 chains.

Take one third of -q- and add it to the shorter east boundary lots 1 or 4, as conditions may require, and thereby determine the length of one of the meridional boundaries of lot 2; to which, again add “one third of “q” and thus obtain the length of the opposite side of lot 2. The areas of lots 1, 2, and 3, in acres, will be found by taking the sum of their respective meridional boundaries, expressed in chains and decimals of a chain.

The area of lot 4 may be had by multiplying its mean width by its mean length.

Finally, to test the entire work, multiply the sum of the latitudinal boundaries by 4, and to the product add the area of the small triangle CAB, if the east boundary is greater than 80.00 chains; but subtract the area of said small triangle if the east boundary is less than 80.00 chains. These operations, correctly performed, will give the true area of the section, which should agree exactly with the total area of its legal subdivisions, obtained in the preceding paragraphs.

Compute areas of lots 5, 6, and 7 of sec 6, as directed previously. Next, write the proportion for “q”.

77.75 : 0.05 :: 60.00 : 0.0386 = q; 1/3 = 0.0129
20.0500 – 0.0386 = 20.01, the E. bdy. of lot 4;
20.0114 + 0.0129 = 20.02, the E. bdy. of lot 3;
20.0243 + 0.0129 = 20.04, the E. bdy. of lot 2.

Then, for the areas of lots 1, 2, 3, and 4, compute thusly:
20.05 + 20.04 = 40.09 acres, the area for lot 1.
20.04 + 20.02 = 40.06 acres, the area for lot 2.
20.02 + 20.01 = 40.03 acres, the area for lot 3.
(20.00 + 20.01)/2 X (17.75 + 17.78)/2 = 35.54 acres, the area for lot 4

The area in acres of a tract 40.00 chains long, adjoining north or west township boundaries (except in NW% section 6), is equal to the sum of its parallel boundaries (expressed in chains and decimals thereof) multiplied by 2.

The area in acres of a tract 60.00 chains long, situated as above described (excluding lot 4, of section 6), may be found by multiplying the sum of its parallel boundaries (expressed in chains and decimals of a chain) by 3.

The area in acres of any section along the north and west boundaries of regular townships (except in section 6) may be had by multiplying the sum of its parallel boundaries (expressed in chains and decimals of a chain) by 4.

Subdivisions closing irregularly to the south or east exterior boundary are to be computed by similar methods.