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Triangle Area From Coordinates (Shoelace Formula)

Enter the coordinates of three vertices to find the triangle area with the shoelace formula. Also shows the perimeter, centroid, and each side length with a coordinate diagram.

Input

Enter the coordinates of the three vertices A, B, and C to compute the triangle area with the shoelace formula, along with the perimeter, centroid, and side lengths.

Vertex A

Vertex B

Vertex C

Result

OGABC

Area

15

Perimeter

17.788289

Centroid

(2.666667, 1.666667)

Side AB

6

Side BC

6.403124

Side CA

5.385165

The area is in squared coordinate units, while the perimeter and side lengths use the same unit as the coordinates.

How it works

  • The area is found with the shoelace formula. For points (x1, y1), (x2, y2), and (x3, y3), area = absolute value of ( x1×(y2−y3) ＋ x2×(y3−y1) ＋ x3×(y1−y2) ) ÷ 2.
  • Because the formula takes the absolute value, the result is the same whether the vertices are listed clockwise or counterclockwise.
  • If the three points are collinear, the area is 0 and they do not form a triangle. Check the coordinates in that case.
  • The perimeter is the sum of the three side lengths. Each side length is the distance between two points, the square root of the sum of the squared coordinate differences.
  • The centroid is the average of the three vertices, with coordinates ((x1＋x2＋x3)÷3, (y1＋y2＋y3)÷3).
  • Coordinate units are arbitrary. The area is in squared coordinate units, while the perimeter and side lengths share the same unit as the coordinates.

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