Question

Formula for finding area of triangle in coordinates are given

Answer 1

We can find Area of triangle using formula 12× base × height if we know the length of base and height of triangle.




We can also find area of triangle using Heron's formula if we know the length of three sides of triangle.

But, how can we find area of triangle if we know only the coordinates of vertices of triangle. If, we know the vertices of triangle then we can definitely use distance formula to find the length of all the sides which can enable us to use Heron's formula to find area of triangle. But, this will become too much lengthy and tedious.

We have a formula which can be directly used on the vertices of triangle to find its area.

If, (x1, x2), (x2, y2) and (x3, y3) are the coordinates of vertices of triangle then
Area of Triangle = 12(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))



Now, we can easily derive this formula using a small diagram shown below.

how to find area of triangle using coordinates of three vertices of triangle



Suppose, we have a △ABC as shown in the diagram and we want to find its area.

Let the coordinates of vertices are (x1, y1), (x2, y2) and (x3, y3).

We draw perpendiculars AP, BQ and CR to x-axis.



Area of △ABC = Area of Trapezium ABQP + Area of Trapezium BCRQ - Area of Trapezium ACRP



⇒ Area of △ABC=12(y1+y2)(x1−x2)
+12(y1+y3)(x3−x1)
−12(y2+y3)(x3−x2)=12(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))
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