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Triangle calculator

This calculator can compute area of the triangle, altitudes of a triangle, medians of a triangle, centroid, circumcenter and orthocenter.

Triangle in coordinate geometry
0 1 2 3 4 5 6 7 8 9 - / . del
Area (default) Medians Altitudes
Centroid (intersection of medians)
Circumcenter (center of circumscribed circle)
Orthocenter (intersection of the altitudes)

Formulas and examples for triangle

Area of the triangle?

The area of a triangle whose vertices are $A(x_A, y_A), B(x_B, y_B)$ and $C(x_C, y_C)$ is given by :

$$ {\color{blue}{ K = \frac12|x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A-y_B)| }} $$

Example:

Find the area of the triangle whose vertices are $A(2, 4), B(3, -1)$ and $C(-3, 3)$.

Solution:

In this example we have: $ x_A = 2,~~ y_A = 4,~~ x_B = 3,~~ y_B = -1, x_C = -3,~~ y_C = 3$. So we have:

triangle area
$$ \begin{aligned} K & = \frac12|x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A-y_B)| \\ K & = \frac12|2(-1 - 3) + 3(3 - 4) -3(4-(-1))| \\ K & = \frac12|2\cdot(-4) + 3\cdot(-1) -3\cdot5| \\ K & = \frac12|-8 - 3 -15| \\ K & = \frac12|-26| \\ K & = \frac12\cdot26 \\ K & = 13 \end{aligned} $$

Centroid of the triangle?

The centroid of a triangle whose vertices are $A(x_A, y_A), B(x_B, y_B)$ and $C(x_C, y_C)$ is given by :

$$ {\color{blue}{ (x,y) = \left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right) }} $$

Example:

Find the centroid of the triangle whose vertices are $A(2, 4), B(3, -1)$ and $C(-3, 3)$.

Solution:

triangle centroid

Using the same $x_A, y_A, x_B, y_B, x_C, y_C$, as in previous example we have:

$$ \begin{aligned} (x,y) & = \left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right) \\ (x,y) & = \left(\frac{2 + 3 - 3}{3}, \frac{4 - 1 + 3}{3}\right) \\ (x,y) & = \left(\frac{2}{3}, 2\right) \\ \end{aligned} $$

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