What Is A Convex Shape?
A convex shape is a shape in which, for every pair of points you pick inside it, the entire straight segment joining them lies inside the shape. Nothing pokes inward. For a convex polygon - a closed figure made of straight sides - this is the same as saying every interior angle is less than $180°$.
The opposite is a concave shape (also called non-convex), which has at least one interior angle greater than $180°$, so it looks like it has a "bite" taken out of it. A line drawn between two points inside a concave shape can pass outside the figure.
The quick test: pick any two points inside the shape and imagine the straight line between them. If that line can ever leave the shape, the shape is concave; if it can never leave, the shape is convex.
Examples Of Convex Shape
These examples build from naming shapes to testing them with the angle rule and the segment rule. Each problem statement is bold; the steps are plain.
Example 1
Is an equilateral triangle a convex shape?
Every triangle has three interior angles that add to $180°$, so no single angle can reach $180°$. Each angle in an equilateral triangle is $60°$.
Since every angle is under $180°$, the triangle is convex.
Final answer: yes, every triangle is convex.
Example 2
A quadrilateral has interior angles $80°$, $200°$, $30°$, and $50°$. Is it convex?
Your first instinct is to check that the angles add to $360°$ (the angle sum of any quadrilateral) and, seeing $80 + 200 + 30 + 50 = 360$, to declare the shape a valid convex quadrilateral. Let's look closer.
The angle sum being correct only confirms it is a genuine quadrilateral. It says nothing about convexity. Scan the individual angles: $200°$ is greater than $180°$.
That single reflex angle means the shape caves in at one vertex, so a segment across it would leave the figure.
Final answer: no, it is concave, because one interior angle ($200°$) exceeds $180°$.
Example 3
Is a circle a convex shape?
A circle is not a polygon, so use the segment test instead of the angle rule. Pick any two points inside a circle and draw the segment between them.
That segment always stays inside the disc; it never bulges out through the curve.
Final answer: yes, a circle (and its filled disc) is convex.
Example 4
Is a regular hexagon convex? Use the interior-angle formula.
The interior angle of a regular polygon with $n$ sides is $\dfrac{(n-2) \times 180°}{n}$. For a hexagon, $n = 6$:
$$\frac{(6-2) \times 180°}{6} = \frac{720°}{6} = 120°$$
Every angle is $120°$, which is under $180°$.
Final answer: yes, a regular hexagon is convex.
Example 5
Which of these are convex: a square, a crescent moon shape, an arrow (chevron)?
A square has four $90°$ angles, all under $180°$ - convex.
A crescent has a curved inward bite, so a segment across the horns leaves the shape - concave.
An arrow / chevron has a notch at the tail, giving one reflex angle - concave.
Final answer: only the square is convex.
Example 6
A polygon has all its diagonals lying inside it. Is it convex or concave?
For a convex polygon, every diagonal lies inside the figure. For a concave polygon, at least one diagonal passes outside.
Since all diagonals lie inside, the shape passes the convex test.
Final answer: it is convex - the "all diagonals inside" property is another way to state convexity.
Why Convexity Matters: "The Tightest Skin Around a Set of Points"
Convexity is one of the oldest useful ideas in geometry because it captures "no dents, no surprises." When you stretch a rubber band around a scattering of nails, it snaps to the convex hull - the smallest convex shape containing them all. That single idea drives a lot of real work.
Structural safety. A convex cross-section spreads load smoothly; a concave notch concentrates stress at the inward corner, which is exactly where cracks start.
Optics and dishes. A convex lens or a satellite dish curves outward so that rays converge predictably, with no inward fold to scatter them.
Algorithms and graphics. Collision detection, packaging, and route-planning lean on convex regions because a straight line between two inside points never escapes - which makes "is it inside?" cheap to check.
Common Mistakes With Convex Shapes
These errors show up the moment a student has to classify a shape under time pressure.
Mistake 1: Judging convexity from a "rounded" look
Where it slips in: Assuming any curvy or smooth-looking shape must be convex.
Don't do this: Calling a crescent moon convex because its edges are smooth curves.
The correct way: Smoothness is not convexity. A crescent has an inward-curving edge, so a segment between its two horns leaves the shape. Apply the segment test, not a gut feeling about how "round" it looks. The rusher who classifies on appearance alone misses inward curves.
Mistake 2: Checking only the angle sum, not each angle
Where it slips in: Confirming the interior angles add to the right total and stopping there.
Don't do this: Seeing a quadrilateral's angles sum to $360°$ and concluding it is convex.
The correct way: The angle sum only confirms the shape is a valid polygon. Convexity needs every individual angle to be under $180°$. Scan for any single reflex angle. The second-guesser who trusts the total without inspecting each angle will pass concave shapes as convex.
Mistake 3: Forgetting non-polygon shapes need the segment test
Where it slips in: Trying to use the "$< 180°$ angle" rule on a shape with curved sides.
Don't do this: Looking for interior angles on a circle or a blob and getting stuck.
The correct way: The angle rule is only for polygons. For any shape - curved or straight - the universal test is the segment rule: can a line between two inside points ever leave the shape? If never, it is convex.
Conclusion
A convex shape is one where every line between two inside points stays inside the shape.
For a convex polygon, this is the same as every interior angle being less than $180°$.
A concave shape has at least one reflex angle (over $180°$) and looks like it has a bite taken out.
All diagonals of a convex polygon lie inside it; some diagonals of a concave polygon lie outside.
Use the angle rule for polygons and the segment test for any shape, including curved ones.
Practise What You Have Learned
Work through these to test your understanding: decide whether a regular octagon is convex using the interior-angle formula; classify a shape with angles $100°$, $95°$, $185°$, $80°$ (Answer to Question 2: concave, because $185° > 180°$); and sketch one convex and one concave hexagon. To take this further with a teacher, explore Bhanzu's geometry tutor, middle school math tutor, or math tutoring. Want to test a shape's convexity by dragging its vertices live? Book a free demo class.
Read More
Types of Polygon - how polygons are grouped, including convex and concave.
Irregular Polygons - where convex and concave shapes both appear.
Geometric Shapes - the wider family of 2D and 3D forms.
Octagon - another convex regular polygon, worked through.
Diagonal of Rectangle - the diagonals of a convex quadrilateral in action.
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