LifeTally

Triangle area from three coordinates

What are your triangle’s area and signed area, and is the point order clockwise, counterclockwise or collinear?

Enter three Cartesian points to calculate two area values and the orientation of their ordered vertices.

Your three points

Use the same perpendicular Cartesian axes, equal axis scale and one consistent coordinate unit. Positive x points right and positive y points up. Enter decimal fractions, not a/b. All six coordinates are required.

Point A: x coordinatePoint A: y coordinatePoint B: x coordinatePoint B: y coordinatePoint C: x coordinatePoint C: y coordinate

How the estimate works

D = (xB − xA)(yC − yA) − (yB − yA)(xC − xA). Signed area is D/2; unsigned area is |D|/2. With the declared axes, positive D means counterclockwise order, negative D clockwise, and zero D collinear or coincident points. Repeated or collinear points are valid with zero area. Classification uses exact differences and products of accepted numerical values, without a near-zero tolerance.

A = (xA, yA); B = (xB, yB); C = (xC, yC)
D = (xB − xA)(yC − yA) − (yB − yA)(xC − xA)
S = D ÷ 2; |S| = |D| ÷ 2

Only three declared real plane points on perpendicular axes with equal scale. This does not calculate geographic, spherical or 3D area, side lengths, angles, perimeter, centroid, graphs, point inclusion or unit conversions. It does not validate measurements, estimate uncertainty or guarantee surveying, physical or safety results. Exact relations of accepted values do not prove relations between measured objects.

OpenStax: cross products and triangle area ↗
Wolfram documentation: positive point orientation ↗

Check the filled example

Our chosen example: A = (−2, 1), B = (3, −1), C = (1, 4). Edge differences are (5, −2) and (3, 3); D = 5×3 − (−2)×3 = 21. Signed area and area are both 10.5 coordinate units²; order is counterclockwise. These are mathematical example values, not standards or a forecast.

Accepted input values

Point A: x coordinate: -2
Point A: y coordinate: 1
Point B: x coordinate: 3
Point B: y coordinate: -1
Point C: x coordinate: 1
Point C: y coordinate: 4

Plan the next step

Use the standard calculator for separate arithmetic. Check the point order, intended axes and coordinate units.

Open the standard calculator →