A GPS receiver, a survey document, or a mapping app gives you a boundary as a list of latitude and longitude points. Turning that list into an area looks like simple geometry, but latitude and longitude are angles on a curved surface, not distances on a flat sheet. This guide explains why the obvious approach goes wrong, then works through the methods that do work: projecting to a local plane and using the shoelace formula, and computing the area directly on a sphere, which is what Land Measure and the tools on this site do.
Why raw degrees don’t work
The shoelace formula (also called the surveyor’s formula) gives the area of any simple polygon from the x and y coordinates of its corners. It is tempting to feed it longitudes as x and latitudes as y. The trouble is that degrees are not a fixed distance:
- One degree of latitude is about 111.2 km everywhere (on a sphere with the mean Earth radius of 6,371 km).
- One degree of longitude is 111.2 km only at the equator. It shrinks with the cosine of the latitude as the meridians converge toward the poles: about 82.6 km at 42° N, and half the equatorial value at 60°.
Take a rectangle near Ames, Iowa, with corners at 42.0300° and 42.0309° N, and −93.6300° and −93.6288° W. On the ground it measures about 99.1 m east to west and 100.1 m north to south, for an area of about 9,919 m². The shoelace formula on the raw degrees gives 0.00000108 “square degrees”, a unit with no fixed size. Multiply that by 111,195 m squared, as if both directions were latitude, and you get 13,353 m², which is 35% too large. The excess is exactly 1 ÷ cos(42.03°), the amount by which the method ignores the shrinking longitude.
Correct methods either convert the angles to meters first or work on the curved surface directly.
Method 1: project to a local plane, then use the shoelace formula
For a parcel a few kilometers across or smaller, you can treat the ground as flat around a reference point and convert each coordinate to meters east (x) and north (y) of it. The simplest way is the equirectangular approximation:
x = R × Δλ × cos φ₀
y = R × Δφ
Here R is the Earth’s radius (6,371,000 m), Δλ and Δφ are the differences in longitude and latitude from the reference point in radians (degrees × π ÷ 180), and φ₀ is a reference latitude, usually the average latitude of the points.
Then apply the shoelace formula to the x and y values:
Area = ½ × | Σ (xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ) |
The sum runs over every corner, and the corner after the last one is the first, which closes the ring.
Worked example: five corners
Here is an irregular five-sided field near the same place, with the corners in order around the boundary. Taking P1 as the origin and φ₀ = 42.03054° (the average latitude), the projected coordinates are:
| Corner | Latitude | Longitude | x, east (m) | y, north (m) |
|---|---|---|---|---|
| P1 | 42.0300 | −93.6300 | 0.00 | 0.00 |
| P2 | 42.0298 | −93.6286 | 115.63 | −22.24 |
| P3 | 42.0307 | −93.6283 | 140.41 | 77.84 |
| P4 | 42.0313 | −93.6292 | 66.08 | 144.55 |
| P5 | 42.0309 | −93.6302 | −16.52 | 100.08 |
Now each shoelace term, xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ:
| Side | Term (m²) |
|---|---|
| P1 → P2 | 0 × (−22.24) − 115.63 × 0 = 0 |
| P2 → P3 | 115.63 × 77.84 − 140.41 × (−22.24) = 12,123.36 |
| P3 → P4 | 140.41 × 144.55 − 66.08 × 77.84 = 15,152.60 |
| P4 → P5 | 66.08 × 100.08 − (−16.52) × 144.55 = 9,001.25 |
| P5 → P1 | (−16.52) × 0 − 0 × 100.08 = 0 |
| Sum | 36,277.21 |
Half the sum is 18,138.6 m², about 1.81 hectares or 4.48 acres. The zero terms come from putting the origin at P1; any origin gives the same total.
Each nonzero term is twice the signed area of a triangle formed by P1 and one side, so the shoelace formula is really adding up the triangles shown in the diagram. A positive sum means the corners run counterclockwise. A negative sum means clockwise, and taking the absolute value handles both.
Limits of the flat approach
The equirectangular approximation is excellent for fields and house lots. It degrades for large areas that span a wide range of latitude, because one cos φ₀ cannot fit every part of the shape, and it fails across the ±180° meridian unless you unwrap the longitudes first.
GIS software often uses a formal projection instead, most commonly UTM (Universal Transverse Mercator), with 6°-wide zones. UTM scales distances by 0.9996 along the middle of each zone and a little above 1 toward its edges, so planar UTM areas can differ from ground areas by up to about two tenths of a percent unless the software corrects for scale.
Method 2: area on a sphere (spherical excess)
The approach used by Land Measure and the tools on this site skips the flat map entirely. It treats the Earth as a sphere with radius R = 6,371,000 m and uses a classic fact of spherical geometry: on a sphere, the angles of a triangle add up to more than 180°, and the excess is proportional to the triangle’s area.
Area = E × R²
where E is the spherical excess in radians: the sum of the polygon’s interior angles minus (n − 2) × 180°, converted to radians. For a polygon of n corners, the method:
- Converts every corner to a point on a unit sphere: (cos φ cos λ, cos φ sin λ, sin φ).
- Splits the polygon into a fan of triangles that share the first corner, exactly like the diagram above.
- Computes each triangle’s excess from the three unit vectors a, b, c, with the formula tan(E/2) = a · (b × c) ÷ (1 + a·b + b·c + c·a). The sign of the triple product makes triangles that fold back over a concave notch count as negative.
- Adds the signed excesses and multiplies the absolute value by R².
Because it works in three dimensions, this method handles concave shapes, very large areas, and boundaries that cross the ±180° meridian, with no projection at all. For the five-corner field it gives 18,138.5 m², within 0.1 m² of the flat result. The two methods only drift apart on large shapes, where curvature starts to matter.
You can reproduce this with the coordinates area calculator by pasting the five latitude and longitude pairs.
Sphere or ellipsoid?
The real Earth is slightly flattened. GPS coordinates are defined on the WGS 84 ellipsoid, and geodesic GIS tools compute areas on it. A sphere of radius 6,371 km is a close stand-in, and the small difference depends on latitude.
For a small parcel, the ellipsoidal area is smaller than the spherical one near the equator (by about 0.45%), almost the same around 35° latitude, and larger toward the poles: about 0.15% larger at 42°, about 0.56% at 60°, and less than 1% even at the poles. For the example field, the ellipsoidal area would be about 28 m² larger than the spherical 18,138.5 m².
Put that next to the measurement itself. If the boundary had been walked 1 m inside its true line all the way round, the area would shrink by roughly the perimeter times 1 m, about 516 m², or nearly 3%. A few meters of phone GPS error at the corners matters far more than the choice between a sphere and an ellipsoid. For legal descriptions, a licensed surveyor works on the appropriate ellipsoid and projection.
Perimeter with the haversine formula
The perimeter is the sum of the distances between consecutive corners, including the closing side from the last corner back to the first. On a sphere, the standard tool is the haversine formula:
h = sin²(Δφ/2) + cos φ₁ × cos φ₂ × sin²(Δλ/2)
d = 2R × atan2(√h, √(1 − h))
For the side from P1 to P2: Δφ = −0.0002° = −0.00000349 rad and Δλ = 0.0014° = 0.0000244 rad. That gives h = 8.54 × 10⁻¹¹ and d = 117.75 m, the same as √(115.63² + 22.24²) from the projected coordinates.
| Side | Length (m) | Bearing |
|---|---|---|
| P1 → P2 | 117.75 | 100.9° |
| P2 → P3 | 103.10 | 13.9° |
| P3 → P4 | 99.88 | 311.9° |
| P4 → P5 | 93.81 | 241.7° |
| P5 → P1 | 101.43 | 170.6° |
| Perimeter | 515.97 (1,692.8 ft) |
Bearings are measured clockwise from true north. The distance between coordinates tool computes both for any pair of points.
Order matters: closing and self-intersection
Every method above assumes the points are listed in order around the boundary, either clockwise or counterclockwise. The direction doesn’t change the area, but the order does.
- Closing the ring. The formulas always join the last point back to the first. Repeating the first point at the end, as GeoJSON and KML require, adds a side of zero length and changes nothing.
- Scrambled order. List the four corners of the Iowa rectangle as SW, SE, NW, NE instead of going round, and the outline becomes a figure eight. Its two halves have opposite signs and cancel, and the “area” comes out at almost exactly zero.
- Self-intersection. Any boundary whose sides cross each other has no single meaningful area. Land Measure and the tools on this site check for crossing sides and flag such shapes so you don’t take the number at face value. If you see that warning, look for two corners entered in the wrong order.
Input precision matters too: five decimal places of a degree are about 1.1 m of latitude, in line with phone GPS. Our guide to coordinate formats covers conversions and common sign and order mistakes.
Putting it into practice
If you already have coordinates, paste them into the coordinates area calculator. To draw a boundary on a satellite map instead, use the land area calculator. In the field, the Land Measure app collects corners by GPS walk, map taps, or typed coordinates, and uses the same spherical formulas shown here.