Concave Polygons: Definition, Properties, Examples
Concave Polygons: Definition, Properties, Examples
A concave polygon is a polygon with at least one reflex interior angle greater than 180°, which makes at least one vertex point inward and at least one diagonal fall outside the shape. This article defines concave polygons, contrasts them with convex ones, lists their properties, and works through examples like the arrowhead and the five-pointed star.
The Dent That Changes Everything About A Shape
Push one corner of a rectangle inward until it caves toward the centre, and the shape stops being convex the instant a single interior angle passes 180°. That one dent is the entire definition of concave, and it changes how the shape's angles behave, where its diagonals land, and even whether a straight line can cross its boundary more than twice.
What is a Concave Polygon?
A concave polygon is a polygon that has at least one interior angle greater than 180° - a reflex angle. Equivalently, a concave polygon is any polygon that is not convex. The reflex angle forces at least one vertex to point into the shape rather than out of it, giving the polygon its characteristic "caved-in" look. The word comes from the same root as cave: something is hollowed inward.
A polygon here means a closed flat figure made of straight sides. Because a triangle's three angles always sum to 180°, no single one can exceed it, so a concave polygon must have at least four sides. That is why every triangle is convex and the smallest possible concave polygon is a quadrilateral like the arrowhead or dart.
How Is A Concave Polygon Different From A Convex One?
This article's companion piece on the convex shape covers the outward-bulging case; here the whole point is the inward dent. The difference is not a matter of degree - a single reflex angle flips a polygon from one category to the other. Three tests separate them:
| Test | Convex polygon | Concave polygon |
|---|---|---|
| Interior angles | Every angle is less than 180° | At least one angle is a reflex angle (>180°) |
| Vertices | All point outward | At least one points inward |
| Diagonals | Every diagonal stays inside | At least one diagonal falls outside |
| A straight cut | A line crosses the boundary at most twice | A line can cross the boundary more than twice |
The diagonal test is the one students find most reliable: draw the diagonals. If any diagonal leaves the shape, the polygon is concave. In a convex polygon, no diagonal ever escapes.
The Properties Of A Concave Polygon
The definition (one reflex angle) forces a small chain of consequences. Here are the properties:
- At least one reflex interior angle. By definition, one angle measures between 180° and 360°.
- At least one inward-pointing vertex. The reflex angle sits at a vertex that caves toward the interior.
- At least one diagonal outside the shape. A diagonal drawn across the dent leaves the polygon's boundary.
- A minimum of four sides. Triangles cannot be concave, so concave polygons start at quadrilaterals.
- A line can meet the boundary more than twice. Convexity guarantees at most two crossings; concavity breaks that guarantee.
The interior angle sum is unchanged. A concave hexagon and a convex hexagon both have interior angles summing to 720°. Concavity redistributes the angles but never changes their total.
Examples Of Concave Polygons
Example 1
Is a five-pointed star a concave polygon? A five-pointed star has ten sides. Between each outer point sits an inner vertex that caves toward the centre, and those inner vertices carry reflex interior angles. Because at least one interior angle exceeds 180°, the star is concave. Final answer: Yes, a five-pointed star is a concave polygon.
Example 2
A student is told an arrowhead is convex "because all four corners are sharp." Is that correct? That reasoning breaks at the notch. The vertex at the back of an arrowhead points inward, and its interior angle is a reflex angle above 180°. The correct method checks the interior angle at every vertex. Final answer: No, the arrowhead is concave, because of the reflex angle at its inward vertex.
Example 3
A quadrilateral has interior angles 60°, 70°, 30°, and 200°. Is it concave? Check whether any angle exceeds 180°. The fourth angle is 200°, which is greater than 180°, so it is a reflex angle. Final answer: Yes, the quadrilateral is concave.
Example 4
A pentagon has four angles of 100° each. Find the fifth angle and state whether the pentagon is concave. The interior angles sum to 540°. Since 140° is less than 180°, and the other four are too, no angle is reflex. Final answer: The fifth angle is 140°, and the pentagon is convex, not concave.
Example 5
Can a regular polygon ever be concave? Final answer: No, a regular polygon is always convex; concavity requires unequal angles with at least one reflex.
Example 6
A plus-sign (cross) shape is drawn as a single 12-sided polygon. How many of its interior angles are reflex? The plus sign has four reflex interior angles, so it is concave.
Where Concave Shapes Show Up: From Pac-Man To Floor Plans
Concave outlines are everywhere.
- Video-game collision. An L-shaped room or a Pac-Man silhouette is concave, and game engines split concave outlines into convex pieces before testing collisions.
- Architecture. An L-shaped or U-shaped floor plan is a concave polygon.
- Computational geometry. The convex hull exists because concave outlines are harder to compute over.
What are the most common mistakes with concave polygons?
Mistake 1: Judging concavity by "how pointy" the corners look
Don't do this: Calling an arrowhead convex because three of its corners are sharp. The correct way: Concavity depends only on whether any interior angle exceeds 180°.
Mistake 2: Assuming a concave polygon has a different angle sum
Don't do this: Recomputing the angle sum with a "penalty" for the reflex angle. The correct way: The interior-angle-sum formula (n−2)×180° holds for concave and convex polygons alike.
Conclusion
- A concave polygon has at least one reflex interior angle, one inward-pointing vertex, and at least one diagonal that falls outside the shape.
- It differs from a convex polygon on three tests: interior angles, vertex direction, and where the diagonals land.
- The smallest concave polygon is a quadrilateral.
Practice these to solidify your understanding
- A quadrilateral has angles 80°, 90°, 40°, and 150°. Is it concave? (Answer: No - no angle exceeds 180°.)
- A hexagon has one angle of 250°. Can it be concave? (Answer: Yes.)
- Draw the diagonals of a five-pointed star and state whether any fall outside. (Answer: Yes, several do.)