Area & partitioning
Area of a PLSS aliquot part reconciled against a measured traverse
The nominal forty of a quarter-quarter set against the area its aliquot corners actually enclose. Build the aliquot corners by proportion from measured section corners, then compute what the description really conveys.
Given
- A retracement traverse recovered four controlling corners of Section 14 and coordinated them on a local grid in US survey feet, northing first:
- South-west section corner N 1000.00 E 1000.00
- South quarter corner N 1002.40 E 3641.30
- Center of section N 3640.50 E 3648.20
- West quarter corner N 3638.90 E 1004.60
- The center of section was set at the intersection of the two center lines and is held as recovered.
- The subject description is 'the NE¼ of the SW¼ of Section 14', nominally 40 acres.
- Aliquot corners are to be established by the standard rule: the center of a quarter section lies at the intersection of straight lines joining the midpoints of its opposite sides, and the quarter-quarter corners lie at the midpoints of the quarter section's sides.
Required
- Coordinates of the four corners of the NE¼ of the SW¼.
- The area that description actually encloses, in acres.
- How that area compares with the nominal 40 acres, and with one quarter of the measured SW¼.
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
Area and dimensions of the measured SW quarter
The SW¼ is the quadrilateral bounded by the south-west section corner, the south quarter corner, the center of section and the west quarter corner. Shoelace it before subdividing: everything downstream is a proportion of this figure, so if it is wrong the quarter-quarter is wrong in the same proportion.
The four sides come out near 2640 ft, which is what a normally executed section gives — a nominal half mile is 2640.00 ft. None of them is exactly 2640.00 ft, and none of them needs to be. The original survey's measurements, not the nominal dimensions, control.
The measured quarter holds 160.057 acres against a nominal 160. The 0.057 acre excess is ordinary: it corresponds to the four sides running an average of about half a foot long.
| Side | From | To | Length (ft) | Bearing |
|---|---|---|---|---|
| South | SW section corner | South ¼ corner | 2641.30 | N 89°56′53″ E |
| East | South ¼ corner | Center of section | 2638.11 | N 0°08′59″ E |
| North | Center of section | West ¼ corner | 2643.60 | S 89°57′55″ W |
| West | West ¼ corner | SW section corner | 2638.90 | S 0°06′00″ W |
| Area | 6,972,093 ft² | 160.057 ac |
Midpoints of the four sides of the quarter
The quarter-quarter corners on the boundary of the SW¼ are the midpoints of its sides — that is what 'aliquot' means here: equal division by proportion, not by measured distance from a fixed end.
A midpoint is the mean of the two end coordinates, taken component by component. Nothing about this depends on the sides being straight, parallel or of equal length, which is the point of the rule: it divides whatever the original survey left, rather than imposing a nominal figure on it.
| Midpoint | Of side | Northing (ft) | Easting (ft) |
|---|---|---|---|
| Mid-south | SW cor – S ¼ cor | 1001.20 | 2320.65 |
| Mid-east | S ¼ cor – Center | 2321.45 | 3644.75 |
| Mid-north | Center – W ¼ cor | 3639.70 | 2326.40 |
| Mid-west | W ¼ cor – SW cor | 2319.45 | 1002.30 |
Center of the quarter section
The center of the SW¼ — which is the south-west corner of the subject forty — is the intersection of the line joining mid-west to mid-east with the line joining mid-south to mid-north. Solve the two-line intersection from the coordinates of the four midpoints.
It is worth noticing that this point is not the average of the four midpoints, and not the average of the four corners of the quarter either. On a quadrilateral whose sides are not parallel, those three constructions give three different points. The aliquot rule specifies the intersection, so the intersection is what is computed.
line 1: mid-west (2319.45, 1002.30) → mid-east (2321.45, 3644.75)
line 2: mid-south (1001.20, 2320.65) → mid-north (3639.70, 2326.40)
intersection: N 2320.45, E 2323.53
this is the center of the SW¼, and the SW corner of the NE¼ SW¼The forty, and what it actually contains
The NE¼ of the SW¼ is bounded by the center of the quarter, the mid-north point, the center of section, and the mid-east point. Shoelace those four in order.
It contains 1,743,267 ft², or 40.020 acres — 867 ft² more than a nominal forty. That surplus is not a blunder to be hunted down. It is the excess in the original survey of Section 14 distributed by the aliquot rule, and it belongs to the owner of this forty as surely as the nominal acreage does.
The description conveys the aliquot part, not forty acres. Where a deed says '40 acres, more or less' the acreage is a description, not a term of the conveyance; the boundary is what the aliquot construction puts on the ground.
| Corner | Description | Northing (ft) | Easting (ft) | Side to next (ft) |
|---|---|---|---|---|
| 1 | Center of SW¼ | 2320.45 | 2323.53 | 1319.25 |
| 2 | Mid-north of SW¼ | 3639.70 | 2326.40 | 1321.80 |
| 3 | Center of section | 3640.50 | 3648.20 | 1319.05 |
| 4 | Mid-east of SW¼ | 2321.45 | 3644.75 | 1321.22 |
| Area | 1,743,267 ft² | 40.020 ac |
Why it is not exactly one quarter of the quarter
One quarter of the measured SW¼ would be 160.057 / 4 = 40.014 acres. The subject forty holds 40.020 acres, which is 0.006 acre more. That is not a rounding artefact.
Cutting a general quadrilateral by the two lines joining opposite side midpoints produces four pieces of unequal area. They are equal only when the figure is a parallelogram, and a section subdivided from a survey with ordinary closure is never quite one. Compute all four and the point is plain: they range from 40.002 to 40.026 acres.
This is why an aliquot area must be computed rather than assumed, and why a description reading 'the NE¼ SW¼, containing 40.014 acres' — one quarter of the measured quarter — would be wrong even though it is derived from measured data.
| Aliquot part | Area (ft²) | Area (acres) |
|---|---|---|
| NE¼ SW¼ (subject) | 1,743,267 | 40.020 |
| NW¼ SW¼ | 1,743,539 | 40.026 |
| SW¼ SW¼ | 1,742,779 | 40.009 |
| SE¼ SW¼ | 1,742,508 | 40.002 |
| Σ | 6,972,093 | 160.057 |
Answer
- Corners of the NE¼ of the SW¼: center of the SW¼ at N 2320.45, E 2323.53; mid-north of the SW¼ at N 3639.70, E 2326.40; center of section at N 3640.50, E 3648.20; mid-east of the SW¼ at N 2321.45, E 3644.75.
- Sides 1319.25, 1321.80, 1319.05 and 1321.22 ft.
- Area = 1,743,267 ft² = 40.020 acres.
- That is 867 ft² (0.020 acre) more than the nominal 40 acres, and 0.006 acre more than one quarter of the measured SW¼ (40.014 acres).
- The whole SW¼ measures 6,972,093 ft² = 160.057 acres against its nominal 160.
Check
Compute all four quarter-quarters of the SW¼ and sum them: 1,743,267 + 1,743,539 + 1,742,779 + 1,742,508 = 6,972,093 ft². That equals the shoelace area of the whole SW¼ exactly, so the center point and the four midpoints are mutually consistent and no piece is double-counted or missed.
Check the aliquot construction against nominal geometry: each side of the forty should be near a nominal quarter mile, 1320.00 ft. The four sides run 1319.05 to 1321.80 ft, all within 2 ft of nominal, which is the right order for a section carrying about 0.4 ft of excess per half mile.
Check the center of the quarter a different way. Its northing, 2320.45, should lie close to the mean of the mid-west and mid-east northings, (2319.45 + 2321.45)/2 = 2320.45, because the mid-west to mid-east line is nearly straight east–west. It does. Its easting, 2323.53, lies close to the mean of the mid-south and mid-north eastings, (2320.65 + 2326.40)/2 = 2323.53. Agreement on both components confirms the intersection was solved and not mis-keyed.
Reasonableness: 40.020 acres on a nominal 40 is an excess of 0.05 percent, matching the 0.036 percent excess found in the whole quarter. A quarter-quarter differing from nominal by several percent would mean a corner was mis-identified, not that the original survey was poor.
More area & partitioning
All in this category- Area of a closed parcel from coordinates by the shoelace ruleA five-corner parcel is given as plane coordinates. Work the cross-product sum term by term to square feet and acres, and use the sign of the result to prove the corner list is in boundary order.
- The same parcel by double meridian distanceThe HAWTHORN TRACT area recomputed on a DMD sheet — the form a plat's computation panel still shows — and set against the coordinate answer to prove the two methods are the same computation.
- Area of a parcel with one circular-arc boundaryA five-sided parcel whose southerly boundary is a circular arc. Compute the area of the chord polygon by coordinates, add the circular segment R²/2·(Δ − sin Δ), and check the segment as sector minus triangle.