Definition:Sign of Area of Triangle/Definition 2

Definition

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

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.

Thus if a person $P$ were to walk around the boundary of $\triangle ABC$ in the direction $AB \to BC \to CA$ where $\triangle ABC$ has a positive area, the interior of $\triangle ABC$ would be on the left of $P$.

Triangle $\triangle ABC$ with Positive Area

Triangle $\triangle ABC$ with Negative Area

Also see

• Equivalence of Definitions of Sign of Area of Triangle

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Coordinates: $7$. Sign of an area