# 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.

 Line $L1: \quad$ $a_1$ = 4 , $b_1$ = 1 , $c_1$ = -25 Line $L2: \quad$ $a_2$ = 2 , $b_2$ = 9 , $c_2$ = 13 Line $L3: \quad$ $a_3$ = -1 , $b_3$ = 4 , $c_3$ = 2 Decimal Places = 2

## 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 .