Area & partitioning
Area from a metes-and-bounds description
A deed of five calls reduced to latitudes and departures, carried to coordinates, and closed out to an area — with the description's own closure tested before the area is trusted.
Given
- A deed describes a parcel by the following calls, beginning at an iron pipe at corner A and running:
- A to B: N 42°30′00″ E, 420.00 ft
- B to C: S 78°15′00″ E, 385.50 ft
- C to D: S 15°40′00″ W, 468.30 ft
- D to E: S 71°05′00″ W, 310.20 ft
- E to A: N 36°59′15″ W, 401.00 ft, to the point of beginning
- Bearings are on the deed's own meridian; distances are horizontal, in US survey feet.
- Corner A is to be held at assumed coordinates N 1000.00, E 1000.00.
Required
- Latitudes and departures for each call, and the closure of the description.
- Coordinates for all five corners.
- The area of the parcel in square feet and acres.
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
Turn the bearings into azimuths
Latitude and departure are D·cos α and D·sin α with α an azimuth measured clockwise from north, so quadrant bearings have to be converted first. The four rules are: NE azimuth = bearing angle; SE azimuth = 180° − angle; SW azimuth = 180° + angle; NW azimuth = 360° − angle.
Converting to azimuths first, rather than applying signs to the sine and cosine of the bearing angle by hand, removes the main source of sign errors in this computation. The trigonometric functions then produce the correct signs without any further thought.
A–B N 42°30′00″ E → α = 42°30′00″
B–C S 78°15′00″ E → α = 180° − 78°15′00″ = 101°45′00″
C–D S 15°40′00″ W → α = 180° + 15°40′00″ = 195°40′00″
D–E S 71°05′00″ W → α = 180° + 71°05′00″ = 251°05′00″
E–A N 36°59′15″ W → α = 360° − 36°59′15″ = 323°00′45″Latitudes and departures
Compute lat = D·cos α and dep = D·sin α for every call and carry three decimals. Three decimals on a 400 ft course is well beyond what was measured, but the closure is going to be a hundredth of a foot and rounding the components to 0.01 ft would bury it.
The columns sum to the closure of the description: how far the last call falls short of, or past, the point of beginning. A deed is not a field traverse and does not have to close, but a deed that does not close well is a deed whose calls have been altered, mis-transcribed, or written from a scaled plat.
| Call | Bearing | Distance (ft) | Azimuth | Latitude (ft) | Departure (ft) |
|---|---|---|---|---|---|
| A–B | N 42°30′00″ E | 420.00 | 42°30′00″ | +309.656 | +283.748 |
| B–C | S 78°15′00″ E | 385.50 | 101°45′00″ | −78.504 | +377.422 |
| C–D | S 15°40′00″ W | 468.30 | 195°40′00″ | −450.902 | −126.460 |
| D–E | S 71°05′00″ W | 310.20 | 251°05′00″ | −100.565 | −293.446 |
| E–A | N 36°59′15″ W | 401.00 | 323°00′45″ | +320.305 | −241.258 |
| Σ | 1985.00 | −0.010 | +0.006 |
Test the closure before computing anything from it
The linear misclosure is the resultant of the two column sums. Compare it with the perimeter to get the relative precision, which is the number a standard is written against.
At about 0.01 ft in 1985.00 ft the description closes at better than one part in 165,000. That is a written description prepared from a computed traverse, not one scaled off a map, and there is nothing to adjust: the misclosure is at the level of the 0.01 ft the coordinates will be rounded to. Where a deed closes at 1:3,000 or worse, the coordinates must be adjusted before an area is taken from them, and the area should carry a stated uncertainty.
ΣLat = −0.010 ft ΣDep = +0.006 ft (from the three-decimal columns above)
e = √( 0.010² + 0.006² ) = √( 0.000100 + 0.000036 ) = √0.000136 = 0.0117 ft
perimeter ΣD = 1985.00 ft
relative precision = 1985.00 / 0.0117 = 1 : 170,200
carried at full precision the closures are −0.0087 and +0.0056 ft, giving e = 0.0103 ft and a precision near 1 : 190,000 — the same verdictRun the coordinates around the parcel
Starting at A (1000.00, 1000.00), add each latitude to the northing and each departure to the easting, rounding each corner to 0.01 ft as it is set. Only four calls are needed — the fifth, E–A, returns to a corner already fixed.
Use the fifth call as a test rather than as a computation. Running E–A from the computed E lands on N 999.995, E 1000.002, which rounds to A's held coordinates. That is the closure of the description reappearing in a form that is easy to read.
| Corner | Northing (ft) | Easting (ft) |
|---|---|---|
| A | 1000.00 | 1000.00 |
| B | 1309.66 | 1283.75 |
| C | 1231.15 | 1661.17 |
| D | 780.25 | 1534.71 |
| E | 679.69 | 1241.26 |
Area by coordinates
With coordinates in hand the area is the ordinary shoelace sum, with the closing course E–A included. The sum is negative, so the calls as written run clockwise — consistent with a description that leaves the point of beginning to the north-east, swings east and south, and returns from the south-west.
Convert to acres against the exact 43,560 ft² and report to 0.001 acre. Reporting a deed area to more decimals than that implies a precision the calls do not carry: 0.001 acre is 43.6 ft², and a corner uncertain by 0.1 ft on a 400 ft boundary already moves the area by more than that.
| Course | Eᵢ · Nᵢ₊₁ | Eᵢ₊₁ · Nᵢ | Difference |
|---|---|---|---|
| A–B | 1000.00 × 1309.66 = 1,309,660.00 | 1283.75 × 1000.00 = 1,283,750.00 | +25,910.00 |
| B–C | 1283.75 × 1231.15 = 1,580,488.81 | 1661.17 × 1309.66 = 2,175,567.90 | −595,079.09 |
| C–D | 1661.17 × 780.25 = 1,296,127.89 | 1534.71 × 1231.15 = 1,889,458.22 | −593,330.32 |
| D–E | 1534.71 × 679.69 = 1,043,127.04 | 1241.26 × 780.25 = 968,493.11 | +74,633.92 |
| E–A | 1241.26 × 1000.00 = 1,241,260.00 | 1000.00 × 679.69 = 679,690.00 | +561,570.00 |
| Σ = 2A | −526,295.49 |
Answer
- The description closes: ΣLat = −0.010 ft, ΣDep = +0.006 ft, linear misclosure 0.0117 ft in a perimeter of 1985.00 ft, or about 1 : 170,000. No adjustment is warranted.
- Coordinates on A (1000.00, 1000.00): B (1309.66, 1283.75), C (1231.15, 1661.17), D (780.25, 1534.71), E (679.69, 1241.26).
- 2A = −526,295.49 ft², so the area is 263,148 ft².
- Area = 6.041 acres (263,147.74 / 43,560 = 6.041041).
Check
Recompute the area on a DMD sheet from the same latitudes and departures. DMDs accumulate 283.75, 944.92, 1195.88, 775.97, 241.26, and the last of these equals the negative of the last departure (−(−241.26) = 241.26) as it must.
The DMD × latitude column is +87,866.03, −74,185.67, −539,222.29, −78,031.54, +77,277.99, summing to −526,295.48 ft². Halved, that is 263,147.74 ft² — the shoelace answer to the hundredth of a square foot.
Inverse the closing line from the computed coordinates: E (679.69, 1241.26) to A (1000.00, 1000.00) gives 401.00 ft on N 36°59′14″ W. The deed calls 401.00 ft on N 36°59′15″ W, so the description is reproduced to the foot-hundredth in distance and to one second in bearing.
Order-of-magnitude check: the parcel is roughly 420 by 470 ft in its longest dimensions, so an area near 200,000 ft² is expected for a shape this irregular. 263,000 ft² is in range; 26,000 or 2,600,000 would say a decimal moved.
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.