Convex Polygon: Definition, Properties, and Examples
Convex Polygon: Definition, Properties, and Examples
TL;DR
A convex polygon is a polygon in which every interior angle is less than 180°, so no corner caves inward and every diagonal stays inside the figure. This article defines the convex polygon, contrasts it with the concave polygon, gives three reliable tests to tell them apart, and works through examples and common mistakes.
A convex polygon is a polygon in which every interior angle measures less than 180°. Equivalently, no part of the boundary caves inward, and every diagonal — every segment joining two corners — lies completely inside the figure. The opposite is a concave polygon (also called non-convex), which has at least one reflex interior angle greater than 180°, producing an inward dent. A triangle is always convex; every regular polygon, such as the hexagon and pentagon, is convex too.
Convex Versus Concave: The Defining Difference
The whole distinction rests on one question: does any corner point inward? If no, convex; if yes, concave. Several consequences follow from that single difference.
| Property | Convex polygon | Concave polygon |
|---|---|---|
| Interior angles | All less than 180° | At least one greater than 180° (reflex) |
| Diagonals | All lie inside the figure | At least one lies partly outside |
| Inward dents | None | At least one |
| Minimum sides | 3 (a triangle) | 4 |
| A line through it crosses the boundary | At most twice | Can cross more than twice |
Notice the consequences cascade from the angle condition. Because no angle is reflex, the boundary never folds back, so diagonals cannot escape and a straight line cannot slice through more than two sides.
The Three Tests For Convexity
Any one of these settles the question; they always agree.
1. The angle test. Measure every interior angle. If all are below 180°, the polygon is convex. A single reflex angle makes it concave. These are the same interior angles studied in interior angles.
2. The diagonal test. Draw the diagonals. If every diagonal lies entirely inside the polygon, it is convex. If even one leaves the figure, it is concave.
3. The line test. Imagine extending each side into a full straight line. If no such line cuts through the interior of the polygon, it is convex. Equivalently, any straight line crosses a convex polygon's boundary in at most two points.
Properties of a Convex Polygon
Convexity guarantees a tidy set of properties that make convex polygons the easy case in geometry and computing.
Every interior angle is less than 180°. This is the definition and the most direct test.
Interior angles still sum to (n−2)×180°. The sum formula holds for any simple polygon, convex or concave, because it depends only on side count.
All diagonals lie inside, and the number of diagonals of an n-gon is ( \frac{n(n-3)}{2} ).
The exterior angles sum to 360°, taken consistently around the figure.
It is the smallest "wrapping" region. The convex hull of a set of points is the tightest convex polygon containing them — the rubber-band shape.
For a regular convex polygon, each interior angle is ( \frac{(n-2)\times 180°}{n} ), which is always below 180°, confirming that every regular polygon is convex.
Examples of Convex Polygon
Example 1
Is a regular pentagon convex? Justify using the angle test.
Each interior angle of a regular pentagon is
[ \frac{(5-2) \times 180°}{5} = \frac{540°}{5} = 108° ]
Every angle equals 108°, which is less than 180°. By the angle test, the regular pentagon is convex.
Example 2
A student says, "this arrowhead shape is convex because it has only four sides, and four-sided shapes are simple." Find the error.
A natural first move is to assume a small side count guarantees convexity. Try the angle test instead of counting sides.
An arrowhead (dart) has one corner that points inward, and the interior angle at that corner is a reflex angle greater than 180°. One reflex angle is enough to make the polygon concave. Side count does not decide convexity — the angles do. The arrowhead is a concave quadrilateral.
Example 3
A quadrilateral has interior angles 80°, 95°, 100°, 85°. Is it convex?
Check each angle against 180°.
[ 80°<180°, , 95°<180°, , 100°<180°, , 85°<180° ]
As a check, the four add to ( 80° + 95° + 100° + 85° = 360° ), the correct sum for a quadrilateral.
Every angle is below 180°, so the quadrilateral is convex.
Example 4
One interior angle of a hexagon measures 210°. Is the hexagon convex or concave, and what does 210° tell you?
An interior angle of 210° is greater than 180°, so it is a reflex angle. A reflex interior angle means the boundary folds inward at that vertex. So the hexagon is concave. The other five angles do not matter — one reflex angle settles it.
Example 5
Use the diagonal test on the capital letter "L" outline (a six-sided shape with one inward corner).
Draw a diagonal connecting the two vertices on either side of the inward corner.
That diagonal passes through the empty notch outside the L's body. Because a diagonal leaves the figure, the L fails the diagonal test. The L is concave — consistent with its 270° reflex angle at the inner corner.
Example 6
An architect lays out a triangulated truss and needs every panel to be convex so loads transfer cleanly along straight members. A panel is a quadrilateral with angles 70°, 110°, 70°, 110°. Does it qualify?
Check each angle: 70°, 110°, 70°, 110° are all below 180°.
Their sum is ( 70° + 110° + 70° + 110° = 360° ), correct for a quadrilateral.
Every angle is below 180°, so the panel is convex and qualifies. Convex panels matter structurally because a load applied inside a convex shape stays supported by the surrounding members.
Where Convexity Earns its Keep: The Easy Case for Computers
Convex polygons are the shapes algorithms love, because convexity removes the awkward edge cases that concave shapes create.
Collision detection in games and robotics. Testing whether two shapes overlap is fast and exact for convex shapes, but concave shapes must first be split into convex pieces. Physics engines almost always store objects as convex polygons or bundles of them.
The convex hull. Given scattered points — GPS readings, pixels, data samples — the smallest convex polygon enclosing them (the convex hull) is a workhorse in pattern recognition, collision bounds, and statistics. It is the rubber-band shape made precise.
Why convexity is the dividing line. The destination is predictability: in a convex region, the straight segment between any two interior points stays inside. That single guarantee is what makes optimisation, shortest-path, and visibility problems tractable.
The idea reaches into higher mathematics as the convex set: a region where the line between any two of its points never leaves it. A convex polygon is simply the polygon version of that property.
Mistakes to Watch For
Mistake 1: Judging convexity by how "neat" a shape looks
Where it slips in: Eyeballing a shape and calling it convex because it looks tidy, without checking angles.
Don't do this: Assume a six-sided shape is convex just because regular hexagons are.
The correct way: Apply a real test — measure the angles, draw the diagonals, or try the line test. The student who trusts a glance will misjudge any shape with a shallow inward dent that the eye glides past.
Mistake 2: Thinking a reflex angle is "just a big angle"
Where it slips in: Recording an interior angle of 200° as if it were ordinary, missing that it signals concavity.
Don't do this: Treat 180° as a soft boundary. Any interior angle strictly above 180° is reflex and forces the polygon to be concave.
The correct way: Flag every interior angle above 180° immediately — even one ends the convexity question. The memoriser who learned "convex = angles small" without anchoring the cutoff at 180° misses borderline reflex angles.
Mistake 3: Believing the interior-angle sum formula only works for convex polygons
Where it slips in: Refusing to apply (n−2)×180° to a concave polygon.
The correct way: The sum formula holds for any simple polygon, because it counts triangles, not angles. A concave hexagon still has interior angles summing to 720°, with one of them just happening to be reflex.
Key Takeaways
A convex polygon has every interior angle less than 180°, so no corner caves inward.
A concave polygon has at least one reflex angle (>180°) and at least one diagonal that leaves the figure.
Three tests agree: the angle test, the diagonal test, and the line test.
The interior-angle sum (n−2)×180° holds for convex and concave simple polygons alike.
Convexity is the case algorithms prefer, from collision detection to the convex hull.
A Practical Next Step
Practice these problems to solidify your understanding. For each, decide convex or concave and name the test you used.
A quadrilateral has angles 60°, 100°, 100°, 100°. Is it convex? (Answer to Question 1: Yes — all angles are below 180°.)
A pentagon has one interior angle of 230°. Convex or concave? (Answer to Question 2: Concave — one reflex angle is enough.)