What this calculator does
This calculator finds the area and perimeter of a triangle directly from the (x, y) coordinates of its three corners, using the shoelace formula. It is the tool to reach for when a triangle is defined by points on a coordinate plane or a map, rather than by side lengths or angles that have already been measured out.
The shoelace formula gets its name from the way the coordinates are cross-multiplied in a criss-cross pattern, like lacing a shoe. It works for any triangle orientation and does not need the vertices listed in any particular order beyond going around the triangle consistently, and the same idea extends to any polygon with more vertices, not just triangles.
The formula
Take the six coordinates of the three vertices and combine them as x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2), then take half the absolute value of that sum. The absolute value matters because the raw sum can come out negative depending on whether the vertices are listed clockwise or anticlockwise; the area itself is never negative. Each side length is found separately with the ordinary distance formula between the two points at its ends.
| Term | Meaning |
|---|---|
| (x1, y1), (x2, y2), (x3, y3) | The coordinates of the three vertices, in any consistent order around the triangle. |
| Shoelace formula | Area = ½ |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|. |
| Collinear points | Three points that lie on a single straight line. They give zero area because no triangle is actually formed. |
The inputs explained
| Field | What to enter |
|---|---|
| Vertex A: x | The x-coordinate of the first vertex. |
| Vertex A: y | The y-coordinate of the first vertex. |
| Vertex B: x | The x-coordinate of the second vertex. |
| Vertex B: y | The y-coordinate of the second vertex. |
| Vertex C: x | The x-coordinate of the third vertex. |
| Vertex C: y | The y-coordinate of the third vertex. |
When to use it
Working from a map or a survey plan
A block of land or a plotted region is often defined by grid coordinates rather than measured side lengths, so calculating area straight from those coordinates skips a step of first working out the sides.
Checking geometry or graphics work
In graphics, mapping and CAD contexts, shapes are frequently stored purely as coordinate lists. The shoelace formula is the standard way to recover an area from that data without redrawing the shape.
Verifying three points actually form a triangle
If the calculated area comes out to zero, the three points are collinear rather than the corners of a triangle, which is a quick sanity check on a coordinate dataset.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is the area for a right triangle at the origin, as the height changes?
A triangle with vertices at (0, 0) and (4, 0), and the third vertex height varying.
| Height of third vertex | Area | Perimeter |
|---|---|---|
| 1 | 2.000 | 9.123 |
| 2 | 4.000 | 10.472 |
| 3 | 6.000 | 12.000 |
| 4 | 8.000 | 13.657 |
| 5 | 10.000 | 15.403 |
| 6 | 12.000 | 17.211 |
How area changes as a vertex moves further from the others
The third vertex moving further along the y-axis from a fixed base.
| y-coordinate of third vertex | Area | Side CA |
|---|---|---|
| 2 | 5.000 | 5.385 |
| 4 | 10.000 | 6.403 |
| 6 | 15.000 | 7.810 |
| 8 | 20.000 | 9.434 |
| 10 | 25.000 | 11.180 |
Questions
Does the order I enter the vertices in matter?
Not for the area or perimeter. The shoelace formula takes the absolute value of the result, so listing the vertices clockwise or anticlockwise gives the same final area either way, as long as each vertex is entered once, consistently.
What if the area comes out as zero?
A zero area means the three points are collinear, lying on a single straight line rather than forming a triangle. Check the coordinates for a typo, such as an accidentally repeated point.
Can I use this for triangles with negative coordinates?
Yes. The shoelace formula and the distance formula both work unchanged with negative x or y values, since only the differences between coordinates matter.
How is this different from the usual triangle area calculator?
A standard triangle area calculator expects side lengths, a base and height, or two sides and an angle. This one starts one step earlier, from raw coordinates, and works out the equivalent side lengths itself.
If you already know the side lengths, the base and height, or two sides and an included angle, use the triangle area calculator instead. To solve a full triangle for all its sides and angles, see the triangle calculator.