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Triangle Area from Coordinates Calculator

Find a triangle's area, perimeter, side lengths, and centroid from its three vertex coordinates using the shoelace formula.

Area
6
Perimeter
12
Side lengths (opposite each vertex, in order)
5, 3, 4
Centroid
(1.3333, 1)

How it works

Given three vertices (x₁,y₁), (x₂,y₂), (x₃,y₃), the shoelace formula gives the enclosed area directly — no need to first find a base and height: area = |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| ÷ 2. The absolute value means the same area comes out whether the points are listed clockwise or counterclockwise.

Each side length comes from the ordinary distance formula applied to a pair of vertices (the Pythagorean theorem on their x and y gap), and the perimeter is the sum of all three. The centroid — the triangle's balance point — is just the average of the three x-coordinates and the average of the three y-coordinates.

The default points, (0,0), (4,0), and (0,3), form the classic 3-4-5 right triangle, so the area comes out to exactly 6. If the three points fall on a single straight line, the shoelace formula evaluates to zero and the calculator reports an error rather than a degenerate 'triangle' with no area.

FAQ

Does the order I enter the points in matter?

Not for the area, perimeter, side lengths, or centroid — all four come out identical regardless of which vertex you label 1, 2, or 3, or whether you list them clockwise or counterclockwise. Only the labeling of which side is 'side A' vs 'side B' vs 'side C' depends on the order, since each is defined as the side opposite a numbered vertex.

What does it mean if the calculator says the points are collinear?

It means all three points sit on one straight line, so no triangle exists — the 'triangle' would have zero width and zero area. Double-check your coordinates; a small typo (like reusing a y-value across all three points along a horizontal line) is the usual cause.

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