How this instrument works
A triangle's perimeter is nothing more than the three edges that close it, added together: P = |AB| + |BC| + |CA|. When a triangle arrives as three vertex coordinates rather than three already-measured side lengths, each edge has to be recovered first — and the tool for that job is the two-point distance formula, itself a direct application of the Pythagorean theorem to the horizontal run and vertical rise between two corners. Run that formula once for each pair of adjacent vertices and add the three results; there is no shortcut around doing it three separate times.
That repetition is the honest shape of the problem, not a limitation of this sheet. A closed boundary is fundamentally three independent lengths, and nothing about one edge reveals another — contrast that with this site's area-from-coordinates sheet, which takes the identical six numbers and reaches a completely different quantity through a completely different route: one signed sum of cross-multiplied coordinates, no square root anywhere, ending in a single absolute value. Same six inputs, two unrelated arithmetic paths, because a boundary's length and the surface it encloses are not the same fact about a shape.
Let the three points slide onto one straight line and the two sheets disagree sharply about what happens next. The enclosed area collapses to exactly zero — there is no longer any surface to measure. The perimeter does not: it settles at precisely twice the distance between the two most extreme points, because the middle point splits that segment in two, and the closed path runs out to it and back, covering the full stretch twice. That degenerate case is also the triangle inequality's equality point — where one side stops being strictly less than the sum of the other two and instead equals it exactly.
- Enter Vertex A's coordinates into the 'Vertex A: x' and 'Vertex A: y' fields.
- Enter Vertex B's coordinates into 'Vertex B: x' and 'Vertex B: y'.
- Enter Vertex C's coordinates into 'Vertex C: x' and 'Vertex C: y', closing the triangle.
- Read Perimeter — the summed length of all three sides, in whatever unit the coordinates use.
- Label the corners in any order you like: with only three vertices, the same three sides always get summed.
Worked example — fencing a right-triangle plot
Stake a triangular plot at A = (0, 0), B = (4, 0), and C = (0, 3), measured in metres. Side AB runs |AB| = √((4−0)² + (0−0)²) = √16 = 4. Side BC runs |BC| = √((0−4)² + (3−0)²) = √(16+9) = √25 = 5. Side CA closes the loop: |CA| = √((0−0)² + (0−3)²) = √9 = 3. Add the three: P = 4 + 5 + 3 = 12 — twelve metres of fence, exactly, because all three side lengths happen to land on whole numbers.
Those are the same three stakes the area-from-coordinates sheet uses for its own example, where they enclose 6 square metres. Here the identical corners report 12 linear metres instead — two true numbers about one plot, reached by unrelated arithmetic, because fencing the boundary and sodding the interior are different jobs. Relabel the corners in reverse order, C then B then A, and the perimeter still comes back as 12: with only three points there is exactly one way to pair them into sides, so no relabeling changes which lengths get added.
Questions
What is the formula for the perimeter of a triangle given its vertices?
Add the three side lengths: P = |AB| + |BC| + |CA|. Because the triangle is given as coordinates rather than pre-measured sides, each length first has to be pulled out with the two-point distance formula — itself a direct application of the Pythagorean theorem to the horizontal and vertical gap between a pair of vertices — and the three results are then summed.
How is this different from finding the same triangle's area from its coordinates?
They start from the identical six numbers and end at unrelated quantities by unrelated routes. Area uses the shoelace formula — cross-multiply and subtract the coordinates, take one absolute value, no square root needed. Perimeter needs the distance formula three separate times, one square root per side, then an addition. A boundary's length and the surface it encloses are two different facts about the same shape.
Does it matter which point I label Vertex A, B, or C?
No. A triangle has only three vertices, so there is exactly one way to pair them into sides — every possible labeling still sums the same three pairwise distances. That is specific to triangles: relabel the corners of a quadrilateral and you can change which sides are treated as adjacent, even crossing the shape into a bowtie. With three points that mistake is not available.
What happens if the three points are collinear rather than forming a real triangle?
The area collapses to exactly zero, but the perimeter does not. It settles at precisely twice the distance between the two most extreme points, because the middle point splits that segment into two pieces that a closed path — A to B to C and back to A — walks over twice, once in each direction. It is also the exact case where the triangle inequality stops being strict and becomes an equality.
What is the easiest way to get this wrong by hand?
Computing two of the three sides and forgetting to close the triangle — adding AB and BC but skipping CA, the edge that returns from the last vertex back to the first. That silently measures an open two-segment path instead of a closed boundary and always under-reports the true perimeter. The other frequent slip is crossing coordinates, pairing one vertex's x with a different vertex's y.
How does the triangle inequality relate to this calculator?
The triangle inequality states that any one side must be shorter than the sum of the other two, or the three points cannot enclose a genuine triangle at all. Three vertex coordinates always produce three real distances, but if those three numbers fail that test — one equalling or exceeding the sum of the other two — the points are collinear rather than a true triangle, and this calculator's perimeter still adds correctly even then.