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Triangle Area from Coordinates Calculator

Area straight from the vertices.

How it is calculated

The tool uses the shoelace (surveyor's) formula on the three vertices: A = |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)| / 2. The absolute value means the vertices can be listed clockwise or anticlockwise and the area is still positive. Coordinates may be negative or decimal, and the area is in the square of the unit used for the coordinates, shown with four decimals. If the three points lie on one line, or two coincide, the triangle is degenerate and the tool shows an area of 0.0000 rather than an error. An empty field gives no result.

Worked example

Take the vertices (1, 1), (7, 2) and (4, 6). First term: 1 × (2 − 6) = −4. Second: 7 × (6 − 1) = 35. Third: 4 × (1 − 2) = −4. The sum is −4 + 35 − 4 = 27. Take the absolute value and halve it: 27 / 2 = 13.5. The calculator shows 13.5000 square units.

Try this example →

Common mistakes

  • Entering side lengths instead of coordinates. This tool needs the (x, y) position of each corner.
  • Mixing the pairs, for example typing x1, x2, y1 in the wrong order. Each vertex needs its own x and y.
  • Expecting a negative area for clockwise order. The absolute value always makes it positive.
  • Mixing units across coordinates, such as x in metres and y in feet.
  • Treating an area of 0 as a mistake in the tool. It means the three points are on one straight line.

About this calculator

When a triangle is defined by three vertex coordinates rather than a base and height, its area can be computed directly with the shoelace formula, also called the surveyor's formula: half the absolute value of the sum of cross-multiplied coordinate pairs, taken in order around the triangle. This avoids the extra step of first computing side lengths or a perpendicular height, working purely from (x, y) positions.

Surveyors use exactly this formula, hence its alternate name, to compute the area of a plot of land from GPS or theodolite-measured boundary coordinates, computer graphics and game engines use it to determine triangle winding order and area for rendering and collision detection, and GIS software uses the same underlying method, generalized to more vertices, for polygon area calculations on maps.

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