Triangle Area from Coordinates Calculator
Find the area of a triangle from the coordinates of its three vertices using the shoelace formula. Enter the (x, y) points for A, B and C, and the calculator returns the area as ½ × |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. It works for any triangle in the plane, however it is oriented.
Formula
Area = ½ × |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
- This is the shoelace (Gauss) formula. It multiplies coordinates crosswise, sums them, halves the result, and takes the absolute value.
- The sign of the un-absolute expression tells you the winding order: positive means the vertices A→B→C go counter-clockwise, negative means clockwise.
- A result of zero means the three points lie on one straight line, so they do not enclose any area.
- The area is in the square of the unit used for the coordinates (coordinates in metres give an area in square metres).
Example: A(0,0), B(6,0), C(0,4)
Inputs
- A — x₁: 0
- A — y₁: 0
- B — x₂: 6
- B — y₂: 0
- C — x₃: 0
- C — y₃: 4
Plugging in gives ½ × |0(0−4) + 6(4−0) + 0(0−0)| = ½ × |24| = 12 square units — matching the ½ × base × height of this right triangle.
Frequently asked questions
What is the shoelace formula?
It computes a polygon's area from its vertex coordinates by cross-multiplying neighbouring points. For a triangle it reduces to ½ × |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|.
Does the order of the vertices matter?
Not for the area — the absolute value removes any sign. The order only affects whether the raw expression is positive (counter-clockwise) or negative (clockwise).
What does an area of zero mean?
It means the three points are collinear (on the same line), so they do not form a real triangle with any enclosed area.
Can I use negative coordinates?
Yes. The formula works for any real coordinates, including negative ones and points in any quadrant.
What units does the area use?
The area is in the square of the coordinate unit. If your x and y values are in centimetres, the area is in square centimetres.