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, and the side-by-side figure above shows all three at once.
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 in a figure with no protractor: 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. Rather than a paragraph, here is the chain as a list, since each property follows from the one before.
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.
One property people expect but do not get: the interior angle sum is unchanged. A concave hexagon and a convex hexagon both have interior angles summing to 720°. Concavity redistributes the angles (one grows past 180°, others shrink to compensate) but never changes their total, a fact that flows from the same types of polygon reasoning used for any n-sided figure.
Examples Of Concave Polygons
Six examples, from spotting concavity by eye to reasoning about the reflex angle numerically.
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?
The intuitive move is to equate "sharp corners" with convex, since sharp angles look small and small angles are below 180°. By that logic the arrowhead's four pointed corners would all be convex.
That reasoning breaks at the notch. The vertex at the back of an arrowhead is not sharp in the outward sense - it points inward, and its interior angle (measured inside the shape) is a reflex angle above 180°. Three corners can look sharp while the fourth caves in.
The correct method checks the interior angle at every vertex, not just how pointed each corner looks. One reflex angle at the notch makes the arrowhead concave.
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.
A single reflex angle is enough.
Final answer: Yes, the quadrilateral is concave. (As a check, $60 + 70 + 30 + 200 = 360°$, the correct sum for a quadrilateral.)
Example 4
A pentagon has four angles of 100° each. Find the fifth angle and state whether the pentagon is concave.
The interior angles of a pentagon sum to $(5-2) \times 180° = 540°$.
Four angles account for $4 \times 100° = 400°$.
Fifth angle $= 540° - 400° = 140°$.
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?
A regular polygon has all interior angles equal. For the polygon to be concave, at least one angle must exceed 180°, which would force every angle to exceed 180°, and that is impossible for a simple closed figure.
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?
Trace the outline of a plus sign. At each of the four inner corners where two arms meet, the boundary caves inward, producing a reflex angle. The eight outer corners are ordinary right angles.
Final answer: The plus sign has four reflex interior angles, so it is concave. Counting the inward corners is the reasoning that identifies concavity in any composite outline.
Where Concave Shapes Show Up: From Pac-Man To Floor Plans
Concave outlines are everywhere once the eye is trained to spot the dent, and they matter in fields where "can a straight line cross this twice or more?" is a real question.
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, because collision math is simpler on convex shapes.
Architecture. An L-shaped or U-shaped floor plan is a concave polygon, and its inward corner is a reflex angle a builder measures directly.
Computational geometry. The convex hull - the tightest convex boundary around a set of points - exists precisely because concave outlines are harder to compute over, so algorithms often wrap them in a convex shell first.
The through-line: the reflex angle that defines concavity is exactly the feature that makes concave shapes computationally awkward, which is why so many algorithms start by breaking them into convex pieces.
What are the most common mistakes with concave polygons?
Two errors dominate, and both come from trusting appearance over the angle measure.
Mistake 1: Judging concavity by "how pointy" the corners look
Where it slips in: On shapes like the arrowhead or star, where sharp outer points draw the eye and the inward notch is easy to miss.
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° - not on how sharp the other corners are. The first instinct is to scan for pointy corners and declare convex; the fix is to hunt for the one inward-caving vertex instead. One reflex angle is decisive.
Mistake 2: Assuming a concave polygon has a different angle sum
Where it slips in: When a student sees a caved-in hexagon and expects its interior angles to total something other than 720°.
Don't do this: Recomputing the angle sum with a "penalty" for the reflex angle.
The correct way: The interior-angle-sum formula $(n-2) \times 180°$ holds for concave and convex polygons alike. Concavity redistributes the angles; it never changes their total. The habit that fixes this is to trust the formula and let the reflex angle be large while the others shrink.
Conclusion
A concave polygon has at least one reflex interior angle (greater than 180°), 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; triangles and regular polygons are always convex.
The interior-angle sum $(n-2) \times 180°$ is the same for concave and convex polygons of the same side count.
Practice these to solidify your understanding
Work through these in order:
A quadrilateral has angles 80°, 90°, 40°, and 150°. Is it concave? (Answer to Question 1: No - no angle exceeds 180°, and they sum to 360°.)
A hexagon has one angle of 250°. Can it be concave? (Answer to Question 2: Yes - that reflex angle alone makes it concave.)
Draw the diagonals of a five-pointed star and state whether any fall outside. (Answer to Question 3: Yes, several do - confirming it is concave.)
If Question 1 gave trouble, return to Example 3 and recheck the reflex-angle test. To keep building polygon reasoning with a teacher, explore Bhanzu's geometry tutor, a middle school math tutor, or math classes online. Want a trainer to walk through concave and convex shapes with your child? Book a free demo class.
Read More
Convex polygon - the strictly-outward polygon where every diagonal stays inside.
Irregular polygons - shapes with unequal sides and angles, where concavity often lives.
Sum of angles in a polygon - the $(n-2)\times 180°$ rule that holds for concave shapes too.
Exterior angles of a polygon - the outside-angle rule that behaves differently at a reflex vertex.
Quadrilaterals - where the smallest concave polygons, like the dart, appear.
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