How to Find the Area of a Triangular Lot From Survey Coordinates

Geometry Lessons

The short answer

A = ½|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|

Put easting in the x slot and northing in the y slot, then run the shoelace formula. Nothing needs measuring on the ground beyond the corners themselves.

Survey data does not arrive as bases and heights. It arrives as coordinates, because that is what a total station records and what a plan stores. Trying to scale a height off a drawing throws away the precision those coordinates already carry, so the sensible move is to compute the area from the corners directly.

Easting is x, northing is y

Coordinates in survey work are usually written northing first, which is the reverse of the x then y order the formula expects. Easting is the horizontal value and belongs in the x slot. Northing is the vertical value and belongs in y.

There is a saving grace here. Swapping the two consistently across all three corners flips the sign of the total but not its size, and since the formula takes the absolute value at the end, the area comes out the same. The genuine danger is swapping them for some corners and not others, which produces a plausible looking number that is simply wrong.

1

List the corners in order

Go around the parcel boundary, either direction. Do not jump across the shape.

2

Put easting in x, northing in y

Consistently for every corner. Mixing the order corner by corner is the error to avoid.

3

Run the shoelace and take the size

Sum the three products, drop any minus sign, then halve.

A worked example

Take a parcel with corners at 0.000 E and 0.000 N, at 42.150 E and 18.640 N, and at 15.320 E and 55.870 N. The first product is 0 times the gap between 18.640 and 55.870, which is zero. The second is 42.150 times 55.870, which is 2354.92. The third is 15.320 times negative 18.640, which is negative 285.56.

Add them and the total is 2069.356. Halve that and the area is 1034.678 square meters. Cross checked against Heron on the three computed side lengths, which are 46.088, 45.890 and 57.932 meters, the answer comes out identical.

A triangular parcel plotted from three survey corners given as easting and northing, with a north arrow, beside the shoelace working giving 1034.678 square meters, 0.1035 hectares and 0.2557 acres.
The coordinates here are local, running from an arbitrary origin at the first corner. Grid coordinates in the hundreds of thousands work identically, though they are harder to check by eye.

Consistency matters more than which order you pick

Swapping easting and northing across every corner costs nothing, because the sign flips and the absolute value removes it. Swapping for one corner and not the others produces a number that looks entirely reasonable and is not the area of anything.

all corners E then Ngivestotal 2069.3561034.678

consistent, so the size of the total is the right one

all corners N then Egivestotal -2069.3561034.678

the sign flips, but the absolute value gives the same area

Grid coordinates and large numbers

Real coordinates are rarely small. A parcel in a provincial grid system might have eastings around 2,540,000 and northings around 7,410,000, and multiplying numbers that size produces intermediate totals with more digits than a calculator display shows comfortably.

Subtract a common offset from every corner before you start. Take the smallest easting off all the eastings and the smallest northing off all the northings, and the shape moves to sit near the origin without changing size or orientation. The area is identical and the arithmetic becomes checkable by eye.

Grid area is not always ground area

Worth knowing before an area gets quoted to anyone. Coordinates on a projected grid carry a scale factor, and on sloping ground the plan area is the flat projection rather than the surface a person would walk. For a small residential lot the difference is usually negligible. Across a large or steep parcel it is not, and which figure is being asked for should be settled before the calculation, not after.

Mistakes that catch people out

  • Mixing the coordinate order between corners. Pick easting first or northing first and hold it for every corner.
  • Listing the corners out of boundary order. Go around the parcel. Jumping across the shape computes the area of a different figure entirely.
  • Rounding coordinates before starting. Millimeter precision in the corners matters more than it looks once the values get multiplied.
  • Treating a negative total as an error. It records the direction you went around. Take the size.
  • Quoting grid area as ground area. On sloping or large parcels these differ, and the distinction matters on a title.

Practice questions

  1. Corners at 0 E 0 N, 30 E 0 N and 0 E 40 N. Find the area in square meters.
  2. The same parcel in hectares.
  3. Corners at 100 E 200 N, 160 E 200 N and 130 E 280 N. Find the area.
  4. A parcel of 4046.856 square meters. How many acres is that?
  5. Corners at 0 E 0 N, 50 E 20 N and 25 E 10 N. Find the area.
Show the answers
  1. half of 30 times 40 gives 600 square meters
  2. 600 divided by 10000 gives 0.06 hectares
  3. shift to 0 E 0 N, 60 E 0 N and 30 E 80 N, then half of 60 times 80 gives 2400 square meters
  4. exactly 1 acre, since that is the definition
  5. the total comes to zero, so the three corners sit on a straight line and enclose nothing

Common questions

How do I find the area of a lot from survey coordinates?

Put easting in the x slot and northing in the y slot, then apply the shoelace formula. Multiply each x by the difference of the other two y values, add the products, take the size and halve it.

Does it matter whether I use northing or easting first?

Only that you are consistent. Swapping them for every corner flips the sign but not the size, so the area is unchanged. Swapping for some corners and not others gives a wrong answer.

Can I use grid coordinates in the hundreds of thousands?

Yes. The formula works on any coordinate values. Subtracting a common offset from every corner first keeps the numbers smaller and easier to check without changing the result.

How do I convert the answer to hectares or acres?

Divide square meters by 10000 for hectares, or by 4046.856 for acres. An area of 1034.678 square meters is 0.1035 hectares or 0.2557 acres.

What if the parcel has more than three corners?

The same method extends to any number of corners. Go around the boundary in order and pair each corner with the next, wrapping the last back to the first.

Should the area match the one on the plan?

Closely, but small differences are normal. Plans may round, may use a different projection, or may report ground area rather than grid area on sloping sites.

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