Area of Triangle

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Theorem

This page gathers a variety of formulas for the area of a triangle.


In Terms of Side and Altitude

The area of a triangle $\triangle ABC$ is given by:

$\dfrac {c \cdot h_c} 2 = \dfrac {b \cdot h_b} 2 = \dfrac {a \cdot h_a} 2$

where:

$a, b, c$ are the sides
$h_a, h_b, h_c$ are the altitudes from $A$, $B$ and $C$ respectively.


In Terms of Two Sides and Angle

The area of a triangle $ABC$ is given by:

$\dfrac 1 2 a b \sin C$

where:

$a, b$ are two of the sides
$C$ is the angle of the vertex opposite its other side $c$.


In Terms of Inradius

Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$.


Then the area $\AA$ of $\triangle ABC$ is given by:

$\AA = r s$

where:

$r$ is the inradius of $\triangle ABC$
$s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.


In Terms of Circumradius

Let $\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$.


Then the area $\AA$ of $\triangle ABC$ is given by:

$\AA = \dfrac {a b c} {4 R}$

where $R$ is the circumradius of $\triangle ABC$.


In Terms of Inradius and Exradii

The area of a $\triangle ABC$ is given by the formula:

$(ABC) = \rho_a \left({s - a}\right) = \rho_b \left({s - b}\right) = \rho_c \left({s - c}\right) = \rho s = \sqrt {\rho_a \rho_b \rho_c \rho}$

where:

$s$ is the semiperimeter
$I$ is the incenter
$\rho$ is the inradius
$I_a, I_b, I_c$ are the excenters
$\rho_a, \rho_b, \rho_c$ are the exradii from $I_a, I_b, I_c$, respectively.


Heron's Formula

Let $\triangle ABC$ be a triangle with sides $a$, $b$ and $c$ opposite vertices $A$, $B$ and $C$ respectively.


Then the area $A$ of $\triangle ABC$ is given by:

$A = \sqrt {s \paren {s - a} \paren {s - b} \paren {s - c} }$

where $s = \dfrac {a + b + c} 2$ is the semiperimeter of $\triangle ABC$.