Triangle Area from Coordinates Calculator
Given the x and y coordinates of three corner points, the area of the triangle they enclose can be found directly with the shoelace formula, a special case of the Gauss area formula for polygons. It evaluates the determinant of the edge vectors, so there is no need to measure side lengths or angles first. This calculator takes the six coordinates of the three vertices and returns the area, the signed (oriented) area, and the perimeter. Use a flat projected coordinate system; for land parcels, project to UTM or state plane first.
Shoelace area formula
signed area = 0.5 * (x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))
area = |signed area|
side a = distance(V1, V2)
side b = distance(V2, V3)
side c = distance(V3, V1)
perimeter = a + b + c
The signed area is positive for counter-clockwise vertex order and negative for clockwise. A zero result means the three points are collinear. Distances use the planar Euclidean formula.
Coordinate area context
- The shoelace formula is exact for any planar triangle, not just right triangles.
- Vertex order changes the sign but not the magnitude of the area.
- Collinear points yield zero area, flagging a degenerate triangle.
- For land parcels, project latitude and longitude to a metric grid (UTM) first.
- The same determinant method extends to any polygon by summing edge cross products.
Triangle area from coordinates: frequently asked questions
What formula computes the area from coordinates?
The shoelace (Gauss) formula: area = 0.5 times the absolute value of x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2). It uses the signed cross product of two edge vectors and works for any triangle given the x and y of its three vertices.
What if the three points are collinear?
If the three points lie on a single straight line, the determinant is zero, so the area is zero. This calculator returns 0.00, which correctly indicates a degenerate triangle with no enclosed area.
Does the order of the vertices matter?
Not for the area, because we take the absolute value. The sign of the determinant tells you orientation (counter-clockwise is positive, clockwise is negative), but the area magnitude is the same regardless of vertex order.
Can I use this for a triangular land parcel?
Yes, if your coordinates are in a flat projected system such as UTM (metres) or state plane (feet). The shoelace formula assumes a flat plane, so it is not suitable for raw latitude and longitude over large areas; project to a metric grid first.
How does this relate to Heron's formula?
Heron's formula computes area from the three side lengths, while the shoelace formula computes it directly from coordinates. Both give the same result. The coordinate method avoids first computing the side lengths and is exact for planar points.
Official sources
- NIST Digital Library of Mathematical Functions: reference for plane geometry.
- U.S. Geological Survey: coordinate systems and areas.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.