Area of a triangle

Summary

The area of a triangle is the measure of the region of the plane that the triangle bounds. It is half the product of a base and the height on that base: $T = \frac{1}{2} b h$. Here $b$ is the length of a side, and $h$ is the distance from the opposite vertex to the line of that side. If two sides have the lengths $a$ and $b$ and the angle $\gamma$ between them, then $T = \frac{1}{2} a b \sin \gamma$. For the vertices $A = (x_A, y_A)$, $B = (x_B, y_B)$ and $C = (x_C, y_C)$ in $\mathbb{R}^2$: $$T = \tfrac{1}{2} \left| (x_B - x_A)(y_C - y_A) - (x_C - x_A)(y_B - y_A) \right|.$$

Remarks

Heron's formula gives the area from the three side lengths alone. In $\mathbb{R}^3$, the area of the triangle $ABC$ is $\frac{1}{2} \|(B - A) \times (C - A)\|$, with the cross product. The coordinate formula without the absolute value is the signed area. It is positive when $A$, $B$, $C$ go round anticlockwise.

Explanations

L1 human

A triangle is half of a parallelogram: two copies of it fit together into one. The parallelogram has the area base times height. So the triangle has half of that, $\frac{1}{2} b h$.