Triangle Interior — Definition, Formula & Examples

The triangle interior is the set of all points that lie inside the three sides of a triangle, not on the sides themselves. It is the enclosed region bounded by the triangle's three line segments.

Given a triangle with vertices $A$ , $B$ , and $C$ , the interior is the open convex region consisting of all points $P$ such that $P$ lies on the same side of each edge as the opposite vertex. Equivalently, $P$ is in the interior if and only if it can be expressed as $P = \alpha A + \beta B + \gamma C$ where $\alpha, \beta, \gamma > 0$ and $\alpha + \beta + \gamma = 1$ .

How It Works

To determine whether a point lies in the interior of a triangle, check that it is strictly inside all three sides — not on any edge and not outside. One practical method is to compute the area of the three smaller triangles formed by connecting the point to each pair of vertices. If the sum of those three areas equals the area of the original triangle, the point is inside or on the boundary. If any of the three smaller areas is zero, the point lies on a side rather than in the interior.

Worked Example

Problem: Triangle ABC has vertices A(0, 0), B(6, 0), and C(3, 4). Determine whether the point P(3, 2) lies in the interior of the triangle.

Step 1: Find the area of triangle ABC using the coordinate formula.

\text{Area}_{ABC} = \tfrac{1}{2}|x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)| = \tfrac{1}{2}|0(0-4) + 6(4-0) + 3(0-0)| = 12

Step 2: Compute the areas of triangles PAB, PBC, and PCA.

\text{Area}_{PAB} = \tfrac{1}{2}|3(0-0) + 0(0-2) + 6(2-0)| = 6$$ $$\text{Area}_{PBC} = \tfrac{1}{2}|3(0-4) + 6(4-2) + 3(2-0)| = 3$$ $$\text{Area}_{PCA} = \tfrac{1}{2}|3(4-0) + 3(0-2) + 0(2-4)| = 3

Step 3: Check whether the three sub-areas sum to the total area and whether all are strictly positive.

6 + 3 + 3 = 12 = \text{Area}_{ABC}

Answer: All three sub-areas are positive and sum to 12, so point P(3, 2) lies in the interior of triangle ABC.

Why It Matters

Knowing whether a point is inside a triangle matters in coordinate geometry proofs, computer graphics (hit detection for triangular regions), and when working with triangle centers like the centroid or incenter, which always lie in the interior.

Common Mistakes

Mistake: Confusing the interior with the boundary. Students sometimes say a point on a side of the triangle is "inside" it.

Correction: Points on the sides belong to the boundary, not the interior. The interior contains only points strictly enclosed by all three sides.