Triangle Vertices Calculator — Area, Sides & Angles from Coordinates
Enter the x and y coordinates of three vertices to compute the triangle's area, all three side lengths, interior angles, perimeter, centroid, and type (equilateral, isosceles, scalene or right).
Computed from the cross-product (Shoelace formula)
- 1
(x₂−x₁)(y₃−y₁)
(4−0) × (3−0) = 12 - 2
(x₃−x₁)(y₂−y₁)
(0−0) × (0−0) = 0 - 3
Cross product
12 − 0 = 12Absolute value taken — sign depends on vertex winding order. - 4
Area = |cross| ÷ 2
|12| ÷ 2 = 6
How does this calculator work?
Given vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃): Area = |(cross-product)| / 2 (Shoelace); sides from Euclidean distance; angles from the law of cosines; centroid = mean of coordinates. Instantly find all triangle properties from three coordinate pairs.
Formula
How this is calculated
The Shoelace formula (also called the surveyor's formula) computes the signed area of any polygon from its vertex coordinates: Area = |((x₂−x₁)(y₃−y₁) − (x₃−x₁)(y₂−y₁))| / 2. The absolute value ensures a positive area regardless of the vertex winding order (clockwise or counter-clockwise).
Each side length is the Euclidean distance between two vertices: for side AB, c = √[(x₂−x₁)² + (y₂−y₁)²]. Once all three sides are known, the interior angles follow from the law of cosines — for angle A opposite side a: cos A = (b² + c² − a²) / (2bc). The centroid (geometric centre) is the mean of the three vertex coordinates: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).
The calculator classifies the triangle by side lengths (equilateral if all equal within floating-point tolerance, isosceles if two match, scalene otherwise) and flags it as a right triangle if the Pythagorean identity holds for the longest side. If the three points are collinear the area is zero and the calculator shows a warning.
Frequently asked questions
The Shoelace formula computes the area of any polygon from its vertex coordinates by taking the difference of products of diagonal vertex pairs. For a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): Area = |(x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂))| / 2, which is equivalent to half the absolute value of the cross product.
The order affects the sign of the cross product but not the magnitude. This calculator takes the absolute value, so the area is the same regardless of whether you enter the vertices clockwise or counter-clockwise. The side labels a, b, c correspond to sides BC, CA and AB respectively.
The centroid (this calculator) is the average of the three vertex coordinates — the balance point where the three medians cross. The circumcenter is the centre of the circle passing through all three vertices. The incenter is the centre of the inscribed circle. All three coincide only for an equilateral triangle.
Also known as
TG we-Calculate Editorial Team. (2026). Triangle Vertices Calculator — Area, Sides & Angles from Coordinates [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/triangle-vertices-calculator
TG we-Calculate Editorial Team. "Triangle Vertices Calculator — Area, Sides & Angles from Coordinates." TG we-Calculate. 2026. https://we-calculate.com/calculator/triangle-vertices-calculator.
TG we-Calculate Editorial Team, "Triangle Vertices Calculator — Area, Sides & Angles from Coordinates," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/triangle-vertices-calculator
@misc{wecalculate_triangle_vertices_calculator, title = {Triangle Vertices Calculator — Area, Sides & Angles from Coordinates}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/triangle-vertices-calculator}}, year = {2026}, note = {TG we-Calculate} }
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