How To Find Area of Triangle Using Determinant Method?

Suppose we have a triangle ABC whose vertices are A( $x_1,y_1$ ), B( $x_2,y_2$ ) and C( $x_3,y_3$ ). In this case, we know the coordinates of the vertices. Hence, we can use a new method to calculate the area of the triangle using the determinant formula rather than the traditional distance formula. IIT JAM exam and NEET exam do not directly ask questions related to the formula. Instead, conceptual questions are asked. Mathematical representation of this formula is given by:

$$\text{Area}=\frac{1}{2}∥\begin{array}{ccc}{\mathrm{|\; x}}_1& y_1& \mathrm{1\; |}\\ {\mathrm{|\; x}}_2& y_2& \mathrm{1\; |}\\ {\mathrm{|\; x}}_3& y_3& \mathrm{1\; |}\end{array}∥$$

The determinant method to calculate the area of a triangle is useful whenever we are given the coordinates of the vertices of a triangle. 

Let us take a look at the derivation of the area of a triangle formula for CUET exam and JEE MAIN entrance exam. There is a triangle with vertices  $$A(x_1,y_1), B(x_2,y_2),\; and\; C(x_3,y_3)$$ First, let us use the shoelace formula to calculate the area of polygon with its vertices $$\text{Area}=\frac{1}{2}\left|\underset{i=1}{\overset{n}{\sum }}\left(x_i{y}_{i+1}\right)-\underset{i=1}{\overset{n}{\sum }}\left(y_i{x}_{i+1}\right)\right|$$ $$$$

$$\text{where}{\mathrm{ x}}_{n+1}=x_1\text{and }{y}_{n+1}=y_1$$ Here n = 3, for a triangle $$\text{Area}=\frac{1}{2}\left|x_1·y_2+x_2·y_3+x_3·y_1-\left(y_1·x_2+y_2·x_3+y_3·x_1\right)\right|$$ Now, the expression within absolute value is written as determinant of matrix:  $$\text{Area}=\frac{1}{2}∥\begin{array}{ccc}{\mathrm{|\; x}}_1& y_1& \mathrm{1\; |}\\ {\mathrm{|\; x}}_2& y_2& \mathrm{1\; |\; <br>}\\ {\mathrm{|\; x}}_3& y_3& \mathrm{1\; |}\end{array}∥$$ Let us now expand this determinant: $$=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$$

The determinant is used to calculate the signed area of the triangle. Sign here indicates the direction i.e. positive means counter clockwise and negative means clockwise. However, we can take the absolute value since the area of a triangle will always be positive.

Let us consider the matrice:

$\left|\begin{array}{ccc}\hfill x_1\hfill & \hfill y_1\hfill & \hfill 1\hfill \\ \hfill x_2\hfill & \hfill y_2\hfill & \hfill 1\hfill \\ \hfill x_3\hfill & \hfill y_3\hfill & \hfill 1\hfill \end{array}\right|$

On expanding the determinant, value of this matrice will be:

x1(y2 – y3) + x2(y3 – y1) + x3(y1-y2)

Hence, area = ½[ x1(y2 – y3) + x2(y3 – y1) + x3(y1-y2)]

Let us take an example to understand the area of a triangle using a determinant method whose vertices are A(2,3), B(5,−1), and C(−3,4). Using the formula $$\text{Area}=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$$ Let us substitute the values: $$x_1=2,y_1$$ So let us now calculate the difference: $$\begin{array}{l}{y}_2-y_3=-1-4=-5\\ y_3-y_1=4-3=1\\ y_1-y_2=3-(-1)=4\end{array}$$ Let us now substitute values and compute the area: $$\begin{array}{l}\text{Area}=\frac{1}{2}\left|2(-5)+5\left(1\right)+(-3)\left(4\right)\right|\\ =\frac{1}{2}\left|-10+5-12\right|\\ =\frac{1}{2}\left|-17\right|\\ =\frac{1}{2}\times 17=8.5\end{array}$$

The determinant method is used for calculating the area of a triangle:

• When you know the coordinates of vertices of a triangle, the determinant method gives a direct formula to compute the area without plotting points or geometric properties such as height and base.
• The determinant formula is systematic while dealing with multiple triangles and automated calculations. 
• Because of the determinant method, there is no need to measure the height and angles of the determinant. 
• A determinant method can even be used for 3D space for finding the area of a triangle which is formed by three points in space.
• Even calculating the volume of a tetrahedron using 4 points is possible using the determinant method.
• The determinant formula is related to cross product of vectors used in vector algebra and physics.
• This determinant method can be used for calculating regardless of the orientation of the triangle.

Let us take a look at the frequently asked questions related to the determinant method for the area of a triangle that are important for entrance exam: