Definition:Sign of Area of Triangle

Definition

Let $\triangle ABC$ be a triangle embedded in the plane.

Definition 1

Let $\theta$ be the angle between $AB$ and $AC$.

If the direction of $\theta$ is anticlockwise, then $\triangle ABC$ is defined as having positive area.

If the direction of $\theta$ is clockwise, then $\triangle ABC$ is defined as having negative area.

Definition 2

Let the sides of $\triangle ABC$ be traversed in the order of its vertices, that is:

$AB \to BC \to CA$

If the area of $\triangle ABC$ is thus described in an anticlockwise direction, then $\triangle ABC$ is defined as having positive area.

If the area of $\triangle ABC$ is thus described in a clockwise direction, then $\triangle ABC$ is defined as having negative area.

Triangle $\triangle ABC$ with Positive Area

Triangle $\triangle ABC$ with Negative Area

Also see

• Equivalence of Definitions of Sign of Area of Triangle

• Results about Sign of Area of Triangle can be found here.