Area of the Triangle Formed by Three co-ordinate Points | Area of the Triangle

How to find the area of the triangle formed by joining the three given points?

(A) In Terms of Rectangular Cartesian Co-ordinates:

Let (x₁, y₁), (x₂, y₂) and (x₃, y₃) be the co-ordinates of the vertices A, B, C respectively of the triangle ABC. We are to find the area of the triangle ABC.


Draw AL, BM and CN perpendiculars from A, B and C respectively on the x-axis.
Then, we have, OL = x₁, OM = x₂, ON = x₃ and AL = y₁, BM = y₂, CN = y₃.
Therefore, LM = OM - OL = x₂ – x₁;
NM = OM - ON = x₂ - x₃;
and LN = ON - OL = x₃ - x₁.

Since the area of a trapezium = 1/2 × the sum of the parallel sides × the perpendicular distance between them,
Hence, the area of the triangle ABC = ∆ABC
= area of the trapezium ALNC + area of the trapezium CNMB - area of the trapezium ALMB
= 1/2 ∙ (AL + NC) . LN + 1/2 ∙ (CN + BM) ∙ NM - 1/2 ∙ (AL + BM).LM
= 1/2 ∙ (y₁ + y₃) (x₃ - x₁) + 1/2 ∙ (y₃ + y₂) (x₂ - x₃) - 1/2 ∙ (y₁ + y₂) (x₂ – x₁)
= 1/2 ∙ [x₁ y₂ – y₁ x₂ + x₂ y₃ - y₂ x₃ + x₃ y₁ – y₃ x₁]
= 1/2 [x₁ (y₂ - y₃) + x₂ (y₃ – y₁) + x₃ (y₁ – y₂)] sq. units.

Also check http://www.mathopenref.com/coordtrianglearea.html
Given the coordinates of the three vertices of a triangle ABC, the area is given by 

 
where A_x and A_y are the x and y coordinates of the point A etc.. 
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