Area of ​​the Triangular Region in relation to the Coordinates of the Vertices

We can determine the area of ​​a triangular region using expressions related to Plane Geometry. In situations involving the position coordinates of the vertices of a triangle, calculations are performed from according to the determinant of a square matrix, formed by the coordinate values ​​of the points of positioning. The constructed matrix must contain in one of its columns the values ​​of the abscissa and in another, the values ​​of the ordinates of the points, a third column will be completed with values ​​equal to 1.


The area of ​​the triangle will be determined by half the value of the determinant. Look:


The vertices of a triangle have the following location coordinates: A(–1, 1), B(4,0) and C(–3, 3). Let's determine the area of ​​this triangular region using the principles of the determinant of a matrix.

Applying Sarrus


main diagonal
(–1) * 0 * 1 = 0
1 * 1 * (–3) = –3
1 * 4 * 3 = 12

Sum: 0 - 3 + 12 = 9

secondary diagonal
1 * 0 * (–3) = 0
(–1) * 1 * (3) = – 3
1 * 4 * 1 = 4

Sum: 0 - 3 + 4 = 1

D = (Sum of the product of the elements of the main diagonal) - (Sum of the product of the elements of the secondary diagonal)

D = 9 - 1
D = 8

A = |D| / two
A = 8 / 2
A = 4

The area of ​​the triangular region with the vertices located at points A(–1, 1), B(4,0) and C(–3, 3) corresponds to 4 area units.


by Mark Noah
Graduated in Mathematics
Brazil School Team

Analytical Geometry - Math - Brazil School

Source: Brazil School - https://brasilescola.uol.com.br/matematica/Area-regiao-triangular-relacao-as-coordenadas-dos-.htm

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