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# Convex Shape - Definition, Properties & Examples

## TL;DR
A convex shape is a shape where the straight line joining any two points inside it stays completely inside - equivalently, every interior angle of a convex polygon is less than 180°. This guide gives the one-line test, the properties, six worked examples, and the mistakes that make students misclassify a shape.

## 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 \( \frac{(n-2) \times 180°}{n} \). 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.

### 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.