How this calculation works
The determinant gives twice the signed area. Averaging the vertex coordinates gives the centroid. Squared side lengths classify the triangle as acute, right or obtuse without rounding a square root. The perimeter sums approximate Euclidean side lengths.
Area = |det(B−A, C−A)|/2; centroid = (A+B+C)/3A worked example
Vertices (0,0), (4,0) and (0,3) give area 6, perimeter 12 and centroid (4/3,1). The triangle is right-angled.
Good to know
Flat Cartesian coordinates with matching units. Each component is between −1,000,000 and 1,000,000; decimal inputs allow up to 6 places and integer fractions have denominators up to 1,000,000. Expressions are not parsed. Exact intermediate fractions have a size limit; reduce unusually complex inputs if that limit is reached. Lengths and angles are decimal approximations.
Reference: MathWorld: shoelace area formula (opens in a new tab) ↗
Vertex order changes the sign, not the physical area
The determinant of B − A and C − A records orientation as well as size. In a conventional Cartesian system with y increasing upward, a positive determinant means counterclockwise vertex order. Reversing B and C changes that sign but preserves the unsigned area.
The outline uses a uniform scale in both coordinate directions. It is a check on the vertices you entered, rather than a separate measurement source. The numerical result uses the original rational coordinates, while the display uses decimal positions.
Three collinear points do not form a nondegenerate triangle. Repeated points also fail that requirement. The calculator reports those cases instead of assigning an ordinary angle classification to a flattened shape.
A centroid is different from a circle center
The triangle centroid is the average of the three vertices. It is where the medians meet and is the balance point of a uniform triangular lamina. For (0,0), (4,0) and (0,3), the centroid is (4/3,1).
The centroid does not generally lie at equal distance from the vertices. That equal-distance point is the circumcenter, which is found by the separate circle-through-three-points calculator. Mixing those centers can give a correct-looking coordinate pair with the wrong geometric meaning.
The angle type is determined from squared side lengths, so the tool does not need to round three inverse-trigonometric calculations first. If the largest squared length equals the sum of the other two, the triangle is right; a larger value makes it obtuse, and a smaller value makes it acute.
A few questions, answered
Does the order of A, B and C matter?
The area, perimeter, centroid and angle type remain the same. Orientation changes when the order is reversed.
Can I enter fractions?
Yes. Use entries such as 1/3, -2 or 0.125, separated by spaces or commas. Do not include brackets or expressions.
Can I use latitude and longitude?
This is a flat Cartesian calculation. Geographic coordinates require a suitable map projection or a geodesic model and should not be treated directly as matching flat units.