Convex Polygon — Definition, Formula & Examples
A convex polygon is a polygon where every interior angle measures less than 180° and any line segment connecting two points inside the polygon stays entirely inside it. Common examples include equilateral triangles, rectangles, and regular pentagons.
A polygon is convex if and only if for every pair of points within the polygon, the line segment joining them lies entirely within the polygon's interior or on its boundary. Equivalently, each interior angle satisfies , and every diagonal is contained within the closed region bounded by the polygon.
Key Formula
Where:
- = Sum of all interior angles of the polygon
- = Number of sides (vertices) of the polygon
How It Works
To check whether a polygon is convex, measure each interior angle — if every angle is strictly less than 180°, the polygon is convex. You can also use the line-segment test: pick any two points inside the shape and draw a straight line between them. If that line never passes outside the polygon, it is convex. Another quick visual check is to look at the vertices: a convex polygon has no "dents" or inward-pointing corners. All regular polygons (equilateral triangle, square, regular pentagon, etc.) are automatically convex because their angles are all equal and always less than 180°.
Worked Example
Problem: A polygon has vertices with interior angles of 90°, 120°, 130°, 110°, and 90°. Determine whether it is convex and verify the angle sum.
Step 1: Check each interior angle against 180°.
Step 2: Since every angle is less than 180°, the polygon is convex.
Step 3: Verify by computing the angle sum. A 5-sided polygon should have an interior angle sum of (5 − 2) × 180° = 540°.
Answer: The polygon is convex, and its interior angles correctly sum to 540°.
Another Example
Problem: A hexagon has interior angles of 100°, 130°, 115°, 200°, 95°, and 80°. Is it convex?
Step 1: Check each angle. The fourth angle is 200°, which exceeds 180°.
Step 2: Because at least one interior angle is greater than 180°, the polygon is not convex — it is concave.
Answer: The hexagon is concave (not convex) because it has a 200° reflex angle.
Visualization
Why It Matters
Convex polygons appear throughout high-school geometry and are essential in proofs about angle sums, tilings, and symmetry. In computer science, convex hulls — the smallest convex polygon enclosing a set of points — are used in collision detection, image processing, and geographic mapping. Understanding convexity also lays the groundwork for optimization problems in calculus and linear programming.
Common Mistakes
Mistake: Confusing "convex" with "regular." Students assume a polygon must have equal sides and equal angles to be convex.
Correction: A convex polygon only requires that every interior angle is less than 180°. The sides and angles do not need to be equal. A rectangle is convex but not regular (unless it is a square).
Mistake: Forgetting that an angle of exactly 180° makes the polygon degenerate, not convex.
Correction: If three consecutive vertices are collinear (forming a 180° angle), the figure is considered degenerate at that vertex. For a polygon to be strictly convex, every interior angle must be strictly less than 180°.