Boundary retracement
Laying out the SW¼ of the NE¼ from section corner coordinates
From four section corner coordinates, protract the quarter corners, fix the center of the section and the center of the north-east quarter, then lay out the SW¼ of the NE¼ and compute its area.
Given
- A normal section in which only the four section corners were recovered; no quarter corners or interior monuments exist and none are called for by the original notes beyond the regular protraction. Coordinates in US survey feet on a local plane grid, to 0.01 ft.
- SW section corner: N 2,014,318.62, E 6,102,455.30.
- SE section corner: N 2,014,296.05, E 6,107,742.88.
- NE section corner: N 2,019,585.40, E 6,107,770.16.
- NW section corner: N 2,019,610.22, E 6,102,480.94.
- Areas are reported in acres to 0.01 acre, with 43,560 square feet to the acre.
Required
- The protracted quarter corners and the center of the section.
- The center of the NE¼ and the four corners of the SW¼ of the NE¼.
- The bearings and lengths of the four sides of that quarter-quarter, and its area.
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
Protract the quarter corners
With no quarter corner monumented and none called for beyond the regular survey, each is protracted at the midpoint of its section line. That is the ordinary rule for the subdivision of a normal section given in the chapter of the BLM Manual of Surveying Instructions (2009) on the subdivision of sections, and it is only available because nothing was found: a recovered original quarter corner would control absolutely, wherever on the line it happened to sit.
The section is not square. Its sides measure 80.12, 80.14, 80.14 and 80.18 chains and none of its bearings is exactly cardinal, so every point below has to be computed rather than assumed.
section lines: south 5,287.63 ft (80.12 ch), east 5,289.42 ft (80.14 ch),
north 5,289.28 ft (80.14 ch), west 5,291.66 ft (80.18 ch)
N¼ = midpoint(NW, NE) = N 2,019,597.81, E 6,105,125.55
E¼ = midpoint(NE, SE) = N 2,016,940.73, E 6,107,756.52
S¼ = midpoint(SE, SW) = N 2,014,307.33, E 6,105,099.09
W¼ = midpoint(SW, NW) = N 2,016,964.42, E 6,102,468.12Center of the section
The center of the section is the intersection of the N¼ to S¼ line with the W¼ to E¼ line. When the quarter corners are protracted at the midpoints of the section sides, that intersection has a convenient property: the two lines bisect each other, so the center is simply the midpoint of either one.
Computing it both ways is therefore free, and it is worth doing, because the property fails the moment a real quarter corner is recovered off the midpoint of its line. In this problem all four quarter corners are protracted, so it holds exactly.
N¼ to S¼: 5,290.54 ft, azimuth 180°17′12″
W¼ to E¼: 5,288.45 ft, azimuth 90°15′24″
midpoint of N¼-S¼ = N 2,016,952.57, E 6,105,112.32
midpoint of W¼-E¼ = N 2,016,952.57, E 6,105,112.32 (identical)
center of section C: N 2,016,952.57, E 6,105,112.32The NE¼ and its own center
The NE¼ is the quadrilateral N¼, NE section corner, E¼, center of section, taken in that order round the boundary. Its area by coordinates is 160.57 acres, a little over nominal, which is consistent with a section running about 0.18 percent long on each side.
To subdivide it, protract a quarter-quarter corner at the midpoint of each of its four sides and take the center of the quarter at the intersection of the lines joining opposite midpoints. Those two lines again bisect each other, so the center is the midpoint of either, and both routes must give the same point.
NE¼ = N¼, NE, E¼, C -> 6,994,501.77 sq ft = 160.57 acres
a = midpoint(N¼, NE) = N 2,019,591.60, E 6,106,447.86 (north side)
b = midpoint(NE, E¼) = N 2,018,263.06, E 6,107,763.34 (east side)
c = midpoint(E¼, C) = N 2,016,946.65, E 6,106,434.42 (south side)
d = midpoint(C, N¼) = N 2,018,275.19, E 6,105,118.94 (west side)
midpoint of a-c = N 2,018,269.13, E 6,106,441.14
midpoint of b-d = N 2,018,269.13, E 6,106,441.14 (identical)
center of the NE¼, Q: N 2,018,269.13, E 6,106,441.14The SW¼ of the NE¼
The south-west quarter-quarter of the NE¼ is bounded by the center of the section at its south-west corner, the midpoint of the quarter's west side, the center of the quarter, and the midpoint of the quarter's south side. Taken in order around the boundary those are C, d, Q and c.
Note which point is which: C is the center of the whole section and is a corner of this forty, while Q is the center of the quarter and is its diagonally opposite corner. Confusing the two is the classic way to lay out the wrong forty by 1,320 ft.
perimeter = 1,322.64 + 1,322.22 + 1,322.50 + 1,322.11 = 5,289.47 ft
each side is close to 20 chains (1,320.00 ft), as a regular forty should be| From | To | Coordinates of the From point (N, E) | Length (ft) | Bearing |
|---|---|---|---|---|
| C (center of section) | d | 2,016,952.57, 6,105,112.32 | 1,322.64 | N 0°17′12″ E |
| d (mid west side) | Q (center of NE¼) | 2,018,275.19, 6,105,118.94 | 1,322.22 | S 89°44′15″ E |
| Q (center of NE¼) | c | 2,018,269.13, 6,106,441.14 | 1,322.50 | S 0°17′28″ W |
| c (mid south side) | C | 2,016,946.65, 6,106,434.42 | 1,322.11 | N 89°44′36″ W |
Area of the forty
Take the four corners in boundary order and compute the area by coordinates. The result is 1,748,654 square feet, or 40.14 acres, which is 0.14 acre over the nominal forty and in proportion with the section's own excess.
A quarter-quarter described as an aliquot part carries whatever area the section actually has; forty acres is the name of the parcel, not a guarantee of its content. A deed that recites 40 acres more or less and describes the SW¼ of the NE¼ conveys this figure, not 40.00 acres.
area by coordinates = 1,748,654 sq ft
1,748,654 / 43,560 = 40.144 acres -> 40.14 acres
excess over nominal = 0.14 acreCheck against the other three forties
Compute the other three quarter-quarters of the NE¼ the same way and require their total to equal the area of the NE¼ found earlier. The four tile the quarter exactly, sharing the four half-lines and meeting at Q, so any discrepancy in the total is an arithmetic error and nothing else.
They agree to three decimal places in acres. The four forties also come out within 0.008 acre of one another, which is the expected signature of a regular protraction from midpoints; a spread of more than a few hundredths would mean a midpoint had been computed from the wrong pair of corners.
NW¼ of the NE¼ = 40.147 acres
NE¼ of the NE¼ = 40.142 acres
SE¼ of the NE¼ = 40.139 acres
SW¼ of the NE¼ = 40.144 acres
sum = 160.572 acres ; NE¼ computed directly = 160.572 acres (agrees)Answer
- Center of section: N 2,016,952.57, E 6,105,112.32
- Center of the NE¼: N 2,018,269.13, E 6,106,441.14
- SW¼ of the NE¼ corners: C 2,016,952.57 / 6,105,112.32; d 2,018,275.19 / 6,105,118.94; Q 2,018,269.13 / 6,106,441.14; c 2,016,946.65 / 6,106,434.42
- Sides: 1,322.64 ft N 0°17′12″ E; 1,322.22 ft S 89°44′15″ E; 1,322.50 ft S 0°17′28″ W; 1,322.11 ft N 89°44′36″ W
- Area = 1,748,654 sq ft = 40.14 acres
Check
The center of the section computed as the midpoint of the N¼ to S¼ line and as the midpoint of the W¼ to E¼ line give the identical point, N 2,016,952.57 E 6,105,112.32, as they must when the quarter corners are protracted at the midpoints of the section sides.
The center of the NE¼ likewise comes out identical from the a-c line and from the b-d line.
The four quarter-quarters of the NE¼ total 160.572 acres, exactly the area of the NE¼ computed directly from its own four corners.
Scale check: the section measures about 80.14 chains on a side, so a forty should run about 0.35 percent over nominal, and 40.14 acres is 0.36 percent over 40.00.
More boundary retracement
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