Area of Triangle in Terms of Side and Altitude

Theorem
The area of a triangle $ABC$ is given by:
 * $\displaystyle \frac {c \cdot h_c} 2 = \frac {b \cdot h_b} 2 = \frac {a \cdot h_a} 2$

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

Proof

 * [[File:Tri.PNG]]

Construct a point $D$ so that $\Box ABDC$ is a parallelogram.

Then we have $\triangle ABC \cong \triangle DCB$, hence their areas are equal.

The area of a parallelogram is equal to the product of one of its bases and the associated altitude.

Thus

where $\left({XYZ}\right)$ is the area of the plane figure $XYZ$.

A similar argument can be used to show that the statement holds for the other sides.

Note
This formula is perhaps the best-known and most useful for determining a triangle's area.

It is usually remembered, and quoted, as half base times height.