How this instrument works
When a triangle's corners are known as plotted coordinates rather than as pre-measured side lengths, its area and perimeter can still be found directly, without an intermediate step of computing each side length by hand first for the area part. The area comes from the shoelace formula, A = ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|, a compact expression built entirely from the six coordinate values. The perimeter still needs each side's length individually, found with the Distance Formula applied three times, once per pair of corners.
This is a genuinely different starting point from Heron's Formula elsewhere on this site, which assumes the three side LENGTHS are already known. Here, the triangle is described by its plotted position instead — the natural form for a triangle defined on a coordinate grid, a map, a CAD drawing, or any system that tracks shapes by their corner points.
The shoelace formula's absolute value strips away a sign that depends only on which rotational direction (clockwise or counterclockwise) the three points happen to be listed in — the enclosed area itself is identical either way, so the formula corrects for that listing-order dependence automatically rather than requiring the corners in one specific order.
- Enter the first corner's coordinates into the Vertex 1: x and Vertex 1: y fields.
- Enter the second corner's coordinates into the Vertex 2: x and Vertex 2: y fields.
- Enter the third corner's coordinates into the Vertex 3: x and Vertex 3: y fields.
- Read Area and Perimeter: the sheet applies the shoelace formula and the Distance Formula simultaneously.
Worked example — corners (0,0), (4,0), (0,3)
A triangle has corners at (0,0), (4,0), and (0,3). Its area is exactly 6 (a right triangle with legs 4 and 3, matching ½×4×3), and its perimeter is exactly 12, the sum of a leg of 4, a leg of 3, and a hypotenuse of 5 — all recovered directly from the three coordinate pairs, with no separate side-length measurement needed.
Three coincident points, all at (0,0), give an area and perimeter of exactly zero — a fully degenerate 'triangle' with no size at all, a boundary case the formulas handle without any special exception needed.
Questions
How do you find a triangle's area from its coordinates?
Use the shoelace formula: A = ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|, built directly from the three corners' coordinate values, with no need to compute the side lengths first.
How is this different from Heron's Formula?
Heron's Formula assumes the three side LENGTHS are already known. This page instead starts from the three corners' COORDINATES, the natural description whenever a triangle is defined by plotted points on a grid, map, or drawing rather than by pre-measured sides.
Does the order of the three vertices matter?
The magnitude of the area doesn't change, but the raw value before taking the absolute value can flip sign depending on whether the three points are listed clockwise or counterclockwise — the absolute value in the formula corrects for that automatically.
What does an area of zero mean?
The three points are collinear (all sitting on the same straight line) or coincide entirely — either way, there's no genuine triangle enclosing any area at all.
How is the perimeter found from coordinates?
By applying the Distance Formula three separate times, once for each pair of adjacent corners, then adding the three resulting lengths together — each individual side length is genuinely needed for the perimeter, unlike the area, which the shoelace formula finds directly.