Triangle Area from Coordinates Calculator — Shoelace Formula
Enter the (x, y) coordinates of three vertices and the calculator finds the triangle area using the shoelace formula. Also reports side lengths, perimeter, centroid and inradius.
Computed by the shoelace (surveyor's) formula
- 1
Shoelace sum
0 × (0 − 4) + 6 × (4 − 0) + 3 × (0 − 0) = 24Signed area from the shoelace formula; sign depends on vertex winding order. - 2
Absolute value
|24| = 24 - 3
Area = |sum| ÷ 2
24 ÷ 2 = 12
How does this calculator work?
Triangle area from three coordinate points: A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. This is the shoelace formula — no angles or heights needed. It also works for any simple polygon. The sign depends on vertex winding order; take the absolute value for the positive area.
Formula
How this is calculated
The shoelace formula (also called the surveyor's formula or Gauss's area formula) computes the area of any simple polygon directly from its vertex coordinates without needing heights, angles or side lengths first. For a triangle with vertices (x₁,y₁), (x₂,y₂) and (x₃,y₃), the signed area is ½ × (x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)). Taking the absolute value removes the sign, which depends only on whether the vertices are listed clockwise or counter-clockwise.
The formula comes from the cross-product interpretation of area: two sides of the triangle can be treated as 2-D vectors, and half the magnitude of their cross product is the triangle area. For multiple vertices the terms are summed in a pattern that resembles interlacing shoelaces — hence the name. The formula extends to any polygon with n vertices.
If the three points are collinear (all on one line) the signed area is zero and no real triangle exists — the calculator returns a warning. The centroid (arithmetic mean of the three vertices) and the inradius (area divided by the semi-perimeter) are also reported as useful derived quantities.
Frequently asked questions
The order affects the sign of the intermediate signed area but not the final absolute area. Going counter-clockwise gives a positive signed area; clockwise gives a negative one. The calculator takes the absolute value, so either ordering gives the same area.
If two or all three vertices coincide, or if all three lie on a straight line, the area is zero. The calculator treats this as a degenerate (non-triangle) case and shows a warning because no real triangle can be formed.
Yes. The shoelace formula works for any finite real coordinates, including negative values and fractional values. Place the vertices anywhere in the plane and the formula gives the correct area.
Also known as
TG we-Calculate Editorial Team. (2026). Triangle Area from Coordinates Calculator — Shoelace Formula [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/area-triangle-coordinates-calculator
TG we-Calculate Editorial Team. "Triangle Area from Coordinates Calculator — Shoelace Formula." TG we-Calculate. 2026. https://we-calculate.com/calculator/area-triangle-coordinates-calculator.
TG we-Calculate Editorial Team, "Triangle Area from Coordinates Calculator — Shoelace Formula," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/area-triangle-coordinates-calculator
@misc{wecalculate_area_triangle_coordinates_calculator, title = {Triangle Area from Coordinates Calculator — Shoelace Formula}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/area-triangle-coordinates-calculator}}, year = {2026}, note = {TG we-Calculate} }
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