# Free Online Calculator for Area of Triangle Formed by Three Lines

An Online calculator to calculate the area of a triangle formed by three lines as shown in the figure below.

## Formulas Used in Calculator

Let the three lines be given by the equations:\( L_1: \quad a_1 x + b_1 y = c_1 \)

\( L_2: \quad a_2 x + b_2 y = c_2 \)

\( L_3: \quad a_3 x + b_3 y = c_3 \)

If any, the point of intersection \( A \) of lines \( L_1 \) and \( L_2 \) is found by solving the systems of equations corresponding to these two lines.

\(\quad a_1 x + b_1 y = c_1 \)

\(\quad a_2 x + b_2 y = c_2 \)

Cramer's rule ( using determinants), gives the \( x \) and \( y \) coordinates of point \( A\) as follows:

\( x_A = \dfrac{ \begin{vmatrix} c_1 & b_1\\ c_2 & b_2 \end{vmatrix} }{\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}} \quad \) , \( \quad y_A = \dfrac{\begin{vmatrix} a_1 & c_1\\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1\\ a_2 & b_2 \end{vmatrix}} \)

Point \( B \) is the intersection of lines \( L_2 \) and \( L_3 \) and its coordinates may be calculated in a similar way as those of point \( A \) above.

\( x_B = \dfrac{ \begin{vmatrix} c_2 & b_2\\ c_3 & b_3 \end{vmatrix} }{\begin{vmatrix} a_2 & b_2\\ a_3 & b_3 \end{vmatrix}} \quad \) , \( \quad y_B = \dfrac{\begin{vmatrix} a_2 & c_2\\ a_3 & c_3 \end{vmatrix}}{\begin{vmatrix} a_2 & b_2\\ a_3 & b_3 \end{vmatrix}} \)

Point \( C \) is the intersection of lines \( L_1 \) and \( L_3 \) and its coordinates may be calculated in a similar way as those of points \( A \) and \( B \) above.

\( x_C = \dfrac{ \begin{vmatrix} c_1 & b_1\\ c_3 & b_3 \end{vmatrix} }{\begin{vmatrix} a_1 & b_1\\ a_3 & b_3 \end{vmatrix}} \quad \) , \( \quad y_C = \dfrac{\begin{vmatrix} a_1 & c_1\\ a_3 & c_3 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1\\ a_3 & b_3 \end{vmatrix}} \)

Once the coordinates have been calculated, we calculate the length of the sides \(AB\), \( BC \) and \( CA \) as follows

\( AB = \sqrt {(x_B - x_A)^2+(y_B - y_A)^2} \) , \( BC = \sqrt {(x_C - x_B)^2+(y_C - y_B)^2} \) , \( CA = \sqrt {(x_A - x_C)^2+(y_A - y_C)^2} \)

We finally use Heron's formula to calculate the area of the triangle as follows:

\( \text{Area} = \sqrt{s(s-AB)(s-BC)(s-CA)} \) , where \( s = \dfrac{1}{2} (AB+BC+CA) \)

## Use of Online Calculator to Find Area of Triangle Formed by Three Lines

Enter the coefficients \( a \),\( b\) and \( c \) as defined above for lines \( L_1\), \( L_2 \) and \( L_3 \) as real numbers and press "Calculate".

The results are: the coordinates of the points of intersections \( A \), \(B \) and \( C \) if any and the area.

## Results

## Activities

Use the calculator to find the area of the triangles formed by the three lines given below.

a)
\( L_1: \quad x = -7 \)
,
\( L_2: \quad x + 5 y = 8 \)
,
\( L_3: \quad - x + 5y = 2\) (Answer: 20 unit squared)

b)
\( L_1: \quad 5x + 6y = -17 \)
,
\( L_2: \quad y = 3 \)
,
\( L_3: \quad - 5x + 4y = -3\) (Answer: 25 unit squared)

c)
\( L_1: \quad - 7x +19 y = -8 \)
,
\( L_2: \quad -3x + 2 y = 15\)
,
\( L_3: \quad x - 15y = -48\) (Answer: 43 unit squared)

## More References and Links

General Equation of a Line: ax + by = c .Cramer's Rule .

Heron's Formula .

Systems of Equations Solver and Calculator .

Equations of Lines in Different Forms .

Online Geometry Calculators and Solvers .