Boundary retracement
Center of section by intersection of the quarter-corner lines
A section is subdivided from eight recovered monuments. The center of section is fixed by intersecting the north-south and east-west quarter-corner lines, and the four quarter-section areas are computed and checked against the whole.
Given
- A retracement of a normal section in which all four section corners and all four quarter corners were recovered as existent original monuments. Coordinates in US survey feet on a local plane grid, to 0.01 ft.
- Section corners: SW at N 512,340.18, E 1,204,662.55; SE at N 512,318.90, E 1,209,946.02; NE at N 517,606.33, E 1,209,972.44; NW at N 517,630.71, E 1,204,684.16.
- Quarter corners: S¼ at N 512,329.02, E 1,207,306.88; E¼ at N 514,960.44, E 1,209,957.86; N¼ at N 517,617.55, E 1,207,330.10; W¼ at N 514,984.60, E 1,204,672.31.
- Areas are to be reported in acres to 0.01 acre, with 43,560 square feet to the acre.
Required
- The two quarter-corner lines, as distance and bearing.
- The coordinates of the center of the section.
- The area of each of the four quarter sections, and a check on their sum.
Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.
Worked solution
The center of section is an intersection, not a midpoint
The center of a normal section is the point where the line joining the north and south quarter corners crosses the line joining the east and west quarter corners. That is the rule given in the chapter of the BLM Manual of Surveying Instructions (2009) on the subdivision of sections, and it is a rule about two lines, not about averaging four points.
The distinction is not academic. Here the midpoint of the north-south quarter-corner line lies at N 514,973.29, E 1,207,318.49, while the midpoint of the east-west line lies at N 514,972.52, E 1,207,315.09. Neither is the center. The true intersection is 0.79 ft south of the first and 3.40 ft east of the second, and a quarter-quarter line run from the wrong point carries that error all the way to the section boundary.
The two quarter-corner lines
Inverse each line. Both run within about sixteen minutes of cardinal, which is normal, and neither is the same length as the other or as any section line: the section is a quadrilateral, not a square, and every dimension has to be measured rather than assumed.
N¼ to S¼: dN = -5,288.53, dE = -23.22, D = 5,288.58 ft, S 0°15′06″ W
W¼ to E¼: dN = -24.16, dE = +5,285.55, D = 5,285.61 ft, S 89°44′17″ E
for reference, the section lines measure:
south 5,283.51 ft (80.05 ch), east 5,287.50 ft (80.11 ch),
north 5,288.34 ft (80.13 ch), west 5,290.57 ft (80.16 ch)Solve the intersection
Write each line in parametric form from its own starting monument and require the two to give the same northing and the same easting. That is two linear equations in the two parameters, and it is worth solving as such rather than reaching for a formula, because the parameters themselves are useful: each one is the fraction of its line at which the crossing occurs.
Both parameters come out very slightly over one half, which is another way of seeing that the center is not the midpoint of either line.
point on line 1: N = 517,617.55 - 5,288.53 t1 , E = 1,207,330.10 - 23.22 t1
point on line 2: N = 514,984.60 - 24.16 t2 , E = 1,204,672.31 + 5,285.55 t2
equate northings: -5,288.53 t1 + 24.16 t2 = -2,632.95
equate eastings: -23.22 t1 - 5,285.55 t2 = -2,657.79
solving: t1 = 0.500148 , t2 = 0.500645
center N = 517,617.55 - 0.500148 x 5,288.53 = 514,972.50
center E = 1,207,330.10 - 0.500148 x 23.22 = 1,207,318.49Confirm the center lies on both lines
Inverse from the center to each of the four quarter corners. The two halves of each quarter-corner line must sum exactly to the full length of that line, which is the arithmetic statement that the point is collinear with both pairs.
N¼ to center = 2,645.08 ; center to S¼ = 2,643.51 ; sum = 5,288.58 (equals the full line)
W¼ to center = 2,646.21 ; center to E¼ = 2,639.40 ; sum = 5,285.61 (equals the full line)Areas of the four quarter sections
Each quarter section is a quadrilateral bounded by two section lines, two quarter-corner half-lines, and it closes on the center. Take the four vertices in boundary order and compute the area by coordinates. No quarter is 160.00 acres, and there is no reason it should be: the section measures a little over 80 chains on every side and the interior monuments are where the original surveyor left them, not where a protractor would put them.
| Quarter | Vertices in order | Area (sq ft) | Area (acres) |
|---|---|---|---|
| NW¼ | NW, N¼, center, W¼ | 7,000,511.56 | 160.71 |
| NE¼ | N¼, NE, E¼, center | 6,986,455.10 | 160.39 |
| SE¼ | center, E¼, SE, S¼ | 6,974,387.76 | 160.11 |
| SW¼ | W¼, center, S¼, SW | 6,994,046.22 | 160.56 |
| Total | - | 27,955,400.65 | 641.77 |
Check the sum against the whole section
The whole section is not the quadrilateral through the four section corners. Its boundary runs through the quarter corners too, and those monuments are not exactly on the straight lines between section corners, so the section is properly an eight-sided figure. Computing the area of that octagon and comparing it with the sum of the four quarters is the check that closes this problem.
The two agree to the last cent of a square foot, which they must: the four quarters tile the octagon exactly, sharing the four half-lines and meeting at the center. Had the straight-sided quadrilateral through the section corners been used instead, it would have come to 27,957,435.42 square feet, or 641.81 acres, and the 2,034.77 square foot difference would have looked like an error in the quarter areas when it is really a statement about where the boundary runs.
octagon SW, S¼, SE, E¼, NE, N¼, NW, W¼ = 27,955,400.65 sq ft = 641.768 acres
sum of the four quarters = 27,955,400.65 sq ft = 641.768 acres
difference = 0.00 sq ftAnswer
- N¼ to S¼ = 5,288.58 ft, S 0°15′06″ W; W¼ to E¼ = 5,285.61 ft, S 89°44′17″ E
- Center of section: N 514,972.50, E 1,207,318.49
- NW¼ = 160.71 acres, NE¼ = 160.39 acres, SE¼ = 160.11 acres, SW¼ = 160.56 acres
- Total 641.77 acres
Check
The center lies on both lines: 2,645.08 + 2,643.51 = 5,288.58 ft equals the N¼ to S¼ length, and 2,646.21 + 2,639.40 = 5,285.61 ft equals the W¼ to E¼ length.
The four quarter areas sum to 27,955,400.65 sq ft, exactly the area of the eight-sided figure through all eight recovered monuments.
Scale check: the section lines measure 80.05, 80.11, 80.13 and 80.16 chains, so a total area a little over 640 acres is expected; 641.77 acres is 0.28 percent over nominal, matching the roughly 0.14 percent excess in each linear dimension.
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