# How to find area of triangle given three vertices?

We can find Area of triangle using formula $\frac{1}{2} \times$ base $\times$ 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 = $\frac{1}{2}(x1(y2-y3)+x2(y3-y1)+x3(y1-y2))$

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

Suppose, we have a $\triangle ABC$ 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 $\triangle ABC$ = Area of Trapezium ABQP + Area of Trapezium BCRQ - Area of Trapezium ACRP

$\Rightarrow$ Area of $\triangle ABC =\frac{1}{2}(y1+y2)(x1-x2)$

$+\frac{1}{2}(y1+y3)(x3-x1)$

$-\frac{1}{2}(y2+y3)(x3-x2) = \frac{1}{2}(x1(y2-y3) + x2(y3-y1) + x3(y1 -y2))$

Example: Find area of triangle whose vertices are (1, 1), (2, 3) and (4, 5)

Solution:

We have (x1, y1) = (1, 1), (x2, y2) = (2, 3) and (x3, y3) = (4, 5)

Using formula:

Area of Triangle = $\frac{1}{2}(x1(y2-y3) + x2(y3-y1) + x3(y1 -y2))$

$=\frac{1}{2}(1(3-5) + 2(5-1) + 4(1-3)) =\frac{1}{2}(-2 + 8 -8) =\frac{1}{2}(-2)=-1$

Because, Area cannot be negative. We only consider the numerical value of answer. Therefore, area of triangle = 1 sq units.

Posted in : Coordinate Geometry, NCERT Solutions, Triangles

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## 2 thoughts on “How to find area of triangle given three vertices?”

1. Jashan Post author

@Arif, this formula works for all 4 quadrants. Take one example and check. You just need to ignore negative sign at the end.

Thanks