Triangle Vertices Calculator

Background.

Three points on a grid fix a triangle completely, and two of the most useful facts about it fall straight out of the coordinates without measuring a single side or angle. The centroid — the point where the triangle would balance on a pin if it were cut from card — is simply the average of the three vertices: add the x-coordinates and divide by three, then do the same for y. It is also where the three medians cross, each running from a vertex to the midpoint of the opposite side, and it sits two-thirds of the way along every median from its vertex.

The area comes from the shoelace formula, named after the criss-cross pattern of products you write down when using it: x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂), halved. Before the absolute value is taken, that bracket carries a sign telling you which way the vertices run — positive anticlockwise, negative clockwise — and the same pattern, extended vertex by vertex, gives the area of any simple polygon. When the three points lie on one straight line it returns exactly zero, which makes it a quick test for collinearity.

Enter the coordinates of A, B and C in any consistent unit and the page returns the centroid and the area in square units. It assumes a flat Cartesian plane, so latitude and longitude need projecting onto a planar grid first.

What is triangle vertices calculator?

The page reports two properties of the triangle defined by three vertices. The centroid is the point whose coordinates are the means of the vertices' x and y values; it is the triangle's geometric centre and the centre of mass of a thin uniform plate of that shape. The area is the region enclosed by the vertices, computed from the coordinates by the shoelace, or determinant, formula and expressed in the square of the coordinate unit.

How to use this calculator.

1. 01Enter the x and y coordinates of vertices A, B and C. The order in which you label the vertices changes neither result.
2. 02Keep all six values in the same unit — metres, feet or grid squares — so the area comes out in square metres, square feet or square grid units.
3. 03For map locations, convert latitude and longitude to a projected grid such as UTM before entering them, because degrees are not a flat distance.
4. 04Read the centroid x and y together as one point, and glance at the area: zero means the three points lie on a line.

The formula.

The centroid is G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3), the arithmetic mean of the vertices. The area is A = |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)| / 2. The bracket equals the cross product of the vectors from one vertex to the other two, which is twice the signed area; its sign records whether the vertices run anticlockwise (positive) or clockwise (negative), and taking the absolute value means the area is never negative. Shifting the whole triangle moves the centroid by the same amount but leaves the area unchanged, and sliding one vertex parallel to the opposite side keeps the area constant because base and height stay the same. Coordinates may be negative or zero, and three collinear points give an area of 0 rather than an error.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.