Triangle Area from Coordinates Calculator
Compute the area of a triangle from the (x, y) coordinates of its three vertices using the shoelace (surveyor’s) formula.
How it works
A = ½ · | x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂) |
This is the shoelace formula specialised to three points: it turns the coordinates directly into an area with no need to find side lengths or heights. The quantity inside the bars is twice the signed area — positive when the vertices are listed counter-clockwise and negative when clockwise — so taking the absolute value and halving gives the true area. If the result is zero, the three points lie on one line.
Worked example
For A(0, 0), B(6, 0) and C(0, 8): cross = 0·(0 − 8) + 6·(8 − 0) + 0·(0 − 0) = 48, so A = |48| / 2 = 24 square units.
Sources: Shoelace formula — Wikipedia
Frequently asked questions
What does a signed area of zero mean?
It means the three points are collinear, so they lie on a single straight line and enclose no area — there is no triangle.
Why is the area negative sometimes before I take the absolute value?
The sign encodes the orientation of the vertices: counter-clockwise order gives a positive signed area and clockwise gives a negative one. The actual area is always the absolute value.
Does this work for any triangle?
Yes. Unlike the base × height method it needs no perpendicular measurement, so it works directly from the coordinates of any triangle in the plane.
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This calculator is for educational purposes. Double-check important results before acting on them.