Updated September 2026
Every number and every line this tool draws comes from four pieces of standard surveying and geometry math, applied in the same order the code runs them. This page walks through each step with the actual formulas, so the plotted shape isn't a black box.
A deed bearing like "N 45°30'00" E" is written in quadrant form: a reference direction (North or South), an angle under 90°, and a second reference direction (East or West). To do vector math, that gets converted to a surveying azimuth — a single angle measured clockwise from North, 0° to 360°. The conversion depends on which quadrant the bearing falls in:
N,E → azimuth = angle
S,E → azimuth = 180 − angle
S,W → azimuth = 180 + angle
N,W → azimuth = 360 − angle
where angle = degrees + minutes/60 + seconds/3600. This is the part of the code that's easiest to get subtly wrong — mixing up which quadrant subtracts from 180 versus 360 will silently mirror or rotate the whole shape while still producing a plausible-looking (but wrong) polygon. It was verified against known test shapes before shipping (see the worked example below).
With the azimuth in hand, each call becomes an (x, y) displacement. This tool plots North as +Y (up) and East as +X (right) — the natural way to read a map — rather than the standard math convention of 0° = +X measured counter-clockwise. That means the trig is:
dx = distance × sin(azimuth)
dy = distance × cos(azimuth)
Sine and cosine are swapped from what you'd expect in a typical math class, precisely because azimuth is measured from North (the y-axis), not from East (the x-axis). Distances entered in meters are converted to feet first (× 3.280839895) so every internal calculation uses one consistent unit.
The point of beginning (POB) is placed at an arbitrary origin (0, 0) — this tool has no way to know the real-world coordinates of the POB unless the deed ties it to a known monument with published coordinates, which is outside this tool's scope. Each call's (dx, dy) is added cumulatively to the running position, in the order the calls are entered, producing the sequence of vertices that gets drawn as the boundary.
Once all vertices are known, the enclosed area uses the shoelace formula (also called the surveyor's formula, which is fitting):
Area = ½ × |Σ(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|
summed over every vertex pair, with the last vertex implicitly connected back to the first. This is exact for any simple (non-self-intersecting) polygon regardless of shape — it doesn't matter whether the lot is a rectangle, an L-shape, or an irregular seven-sided parcel; the formula gives the true enclosed area from the vertex coordinates alone. The result is reported in square feet and converted to acres by dividing by 43,560.
If the calls were transcribed correctly, the cumulative sum of every (dx, dy) should return exactly to (0, 0). It almost never does exactly, because of field survey rounding, and the raw distance between the final plotted point and the POB is the closure error. The tool also reports this as a precision ratio — perimeter divided by closure error, expressed like "1:12,500" — which is the standard way surveyors communicate how tight a traverse closes, because a 2-foot error means something very different on a 200-foot lot versus a 20,000-foot rural tract. Full detail on interpreting that ratio is on the closure error page.
Before shipping, the math was checked against a shape with a known, exact answer. Four cardinal-direction calls — due north 50 ft, due east 100 ft, due south 50 ft, due west 100 ft — describe a plain 100×50 rectangle. Walking them through the exact formulas above lands back on (0, 0) to within floating-point rounding (about 10⁻¹⁴ ft, not a real error), and the shoelace formula returns exactly 5,000 sq ft — matching 100 × 50 computed the ordinary way. A second check using all four quadrants (a 45°-rotated square, i.e. a diamond, built from N45°E, S45°E, S45°W, and N45°W calls of equal length) also closed exactly and returned the expected area, confirming all four quadrant-to-azimuth branches are correct, not just the simplest case.
This is planar (flat-earth) geometry on an arbitrary coordinate system — it doesn't correct for a survey datum, doesn't account for the earth's curvature (irrelevant at the scale of almost any single parcel, but real over miles), doesn't distinguish magnetic north from true north or apply a magnetic declination correction, and doesn't currently support curved calls (arcs described by a radius and central angle) — only straight-line bearing-and-distance calls. A licensed surveyor's actual plat accounts for all of that; this tool is a visualization of the geometry a set of calls describes, not a substitute for that work.