Concave Polygon — Definition, Formula & Examples
A concave polygon is a polygon that has at least one interior angle greater than 180°. This causes part of the shape to "cave in," as if a vertex were pushed inward.
A polygon is concave if and only if there exists at least one interior angle whose measure exceeds 180° (a reflex angle), or equivalently, if a line segment connecting two points inside the polygon can pass outside the polygon's boundary.
How It Works
To check whether a polygon is concave, measure each interior angle. If any angle is greater than 180°, the polygon is concave. 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 ever passes outside the polygon, the shape is concave. A concave polygon always looks like it has a "dent" or indentation on at least one side.
Worked Example
Problem: A quadrilateral has interior angles of 50°, 80°, 210°, and 20°. Is it concave or convex?
Step 1: Check that the angles sum to 360°, as required for any quadrilateral.
Step 2: Look for any interior angle greater than 180°. The angle of 210° is a reflex angle, exceeding 180°.
Answer: The quadrilateral is a concave polygon because it contains a reflex interior angle of 210°.
Why It Matters
Concave polygons appear in real-world shapes like arrow-heads, star outlines, and floor plans with indentations. Recognizing concavity matters when calculating area, because standard formulas for regular polygons assume convexity, so concave shapes often need to be split into simpler convex pieces first.
Common Mistakes
Mistake: Confusing concave and convex by forgetting which word means "caved in."
Correction: Remember that conCAVE has the word "cave" in it — the shape has a cave-like dent where a vertex points inward.