Convex vs Non-Convex Polygons: Complete Guide & Examples

A convex polygon has all interior angles less than 180° with no indentations, while a non-convex (concave) polygon has at least one interior angle greater than 180°, creating an indentation or « cave » in the shape. Every triangle and every regular polygon is convex; stars, arrows, and L-shapes are classic non-convex examples.

Still finding it hard to tell whether a polygon « pokes out » or « caves in »? This comprehensive guide walks through how to identify, classify, and work with both types.

What Is a Polygon?

Before diving into convex vs non-convex, let’s establish what a polygon is:

Polygon Definition: A closed, two-dimensional shape made of straight line segments that connect to form a closed figure.

Key characteristics of ALL polygons:

• Made of straight sides (not curves)
• Sides connect at vertices (corner points)
• Must be closed (no gaps)
• Has at least 3 sides

Examples of polygons: Triangle, square, pentagon, hexagon, octagon

NOT polygons: Circle (curved), open shape (not closed), shape with curves

What Is a Convex Polygon?

A convex polygon is a polygon where ALL interior angles are less than 180° and the shape « bulges outward » with no indentations.

📐 Convex Polygon Definition:
A polygon is convex if:
• ALL interior angles are less than 180°
• NO indentations or « caves » in the shape
• ALL diagonals lie inside the polygon
• Any line segment connecting two points inside stays completely inside

Think of it as: The polygon « pokes out » everywhere – like a stretched rubber band around pins. There are no dents pushing inward.

What Is the Line Test for Convex Polygons?

The Line Test: Draw a line through any side of the polygon and extend it. In a convex polygon, the entire shape stays on one side of that line.

Examples of convex polygons:

• All regular polygons (equilateral triangle, square, regular pentagon, regular hexagon, etc.)
• Rectangles and parallelograms
• Any triangle (all triangles are convex!)
• Trapezoids (most types)

What Is a Non-Convex (Concave) Polygon?

A non-convex polygon (also called a concave polygon) is a polygon that has at least ONE interior angle greater than 180°, creating an indentation that « caves in. »

📐 Non-Convex (Concave) Polygon Definition:
A polygon is non-convex (concave) if:
• At least ONE interior angle is greater than 180° (called a reflex angle)
• Has at least one indentation or « cave »
• At least one diagonal goes outside the polygon
• Some line segments connecting two interior points pass outside the shape

Think of it as: The polygon has a « dent » or « bite taken out of it » – like a star or an arrow pointing inward.

What Is the Line Test for Non-Convex Polygons?

The Line Test: Draw a line through any side and extend it. In a non-convex polygon, the line will pass through the interior of the shape, dividing it into parts on both sides of the line.

Examples of non-convex (concave) polygons:

• Star shapes (like a 5-pointed star)
• Crescent shapes
• Arrow shapes
• L-shapes or C-shapes
• Any polygon with an indentation

What Are the Key Differences Between Convex and Non-Convex Polygons?

[IMAGE: Convex and non-convex polygon examples with labeled angles and diagonals – visual comparison showing interior angles reflex angles and diagonal placement – geometry diagram for students]

How Do You Test If a Polygon Is Convex or Non-Convex?

There are several methods to determine if a polygon is convex or non-convex:

Method 1: The Interior Angle Test

Steps:

1. Measure ALL interior angles of the polygon
2. If ALL angles are less than 180°, it’s convex
3. If ANY angle is 180° or more, it’s non-convex

Example: A pentagon with angles 108°, 120°, 95°, 110°, 107° → All < 180° → CONVEX

Example: A pentagon with angles 80°, 100°, 90°, 85°, 245° → One > 180° → NON-CONVEX

Method 2: The Diagonal Test

Steps:

1. Draw ALL possible diagonals (lines connecting non-adjacent vertices)
2. If ALL diagonals stay INSIDE the polygon, it’s convex
3. If ANY diagonal goes OUTSIDE, it’s non-convex

Why this works: In a convex polygon, there are no indentations for diagonals to escape through. In a non-convex polygon, the « dent » causes some diagonals to pass outside.

Method 3: The Extended Side Test

Steps:

1. Pick any side of the polygon
2. Extend that side as a straight line in both directions
3. Check if the entire polygon stays on ONE side of the line
4. If yes for ALL sides → CONVEX
5. If any side cuts through → NON-CONVEX

Method 4: The Visual « Dent » Check

Simplest method: Just look at the shape!

• Does it have any indentations, caves, or dents? → NON-CONVEX
• Does it bulge outward everywhere? → CONVEX

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What Are Some Special Cases Worth Knowing?

All Triangles Are Convex

Key fact: EVERY triangle is convex, with NO exceptions!

Why? The sum of interior angles in any triangle is always 180°. Since there are 3 angles, each must be less than 180° for the sum to work. Therefore, no triangle can have a reflex angle (> 180°), making all triangles convex.

Types of triangles (all convex):

• Equilateral triangles
• Isosceles triangles
• Scalene triangles
• Right triangles
• Acute triangles
• Obtuse triangles

All Regular Polygons Are Convex

A regular polygon has all sides equal and all angles equal. Regular polygons are ALWAYS convex.

Regular polygons (all convex):

• Equilateral triangle (3 sides)
• Square (4 sides)
• Regular pentagon (5 sides)
• Regular hexagon (6 sides)
• Regular octagon (8 sides)
• Regular decagon (10 sides)

Why? In a regular polygon, all interior angles are equal. For the polygon to be regular, these angles must be less than 180°, making it automatically convex.

Quadrilaterals Can Be Either

Convex quadrilaterals:

• Squares
• Rectangles
• Parallelograms
• Trapezoids
• Rhombuses
• Kites (most)

Non-convex quadrilaterals:

• Dart/Arrowhead shapes
• Bowtie shapes (self-intersecting)
• Some kites

Where Do You See Convex and Non-Convex Polygons in Real Life?

Convex Polygons in Real Life

Stop signs: Regular octagons (8 sides) – convex, like the one at any four-way intersection in Ahuntsic

Soccer balls (pentagons): Regular pentagons – convex

Honeycomb cells: Regular hexagons – convex

Picture frames: Rectangles – convex

Pizza slice: Triangle – convex

Non-Convex Polygons in Real Life

Stars: 5-pointed stars – non-convex (concave)

Crescent moon: Crescent shape – non-convex

Arrow pointing: Arrow shapes – non-convex

Letter shapes: L, C, V (some letters) – non-convex, like the L-shaped floor plan common in duplex conversions across Verdun

Cookie with bite: Indented cookie – non-convex

Practice Problems

Basic Identification Problems

1. Problem 1: Is a square convex or non-convex?
   Solution: CONVEX
   All interior angles = 90° (all < 180°), no indentations, all diagonals stay inside.
   Check: ✓
2. Problem 2: Is a 5-pointed star convex or non-convex?
   Solution: NON-CONVEX (concave)
   Has indentations at each point, some angles > 180° (reflex angles).
   Check: ✓
3. Problem 3: Can any triangle be non-convex?
   Solution: NO
   ALL triangles are convex. Sum of angles = 180°, so each angle must be < 180°.
   Check: ✓
4. Problem 4: Is a regular hexagon convex or non-convex?
   Solution: CONVEX
   All regular polygons are convex. Each interior angle = 120° (< 180°).
   Check: ✓
5. Problem 5: Is the letter « L » shape convex or non-convex?
   Solution: NON-CONVEX (concave)
   Has an indentation where the two bars meet, creating a reflex angle > 180°.
   Check: ✓

Intermediate Problems

6. Problem 6: A quadrilateral has angles 85°, 95°, 90°, and 90°. Is it convex or non-convex?
   Solution: CONVEX
   All angles < 180°. Check: 85° + 95° + 90° + 90° = 360° ✓ (sum of quadrilateral angles)
   Answer: CONVEX
7. Problem 7: A pentagon has 4 angles of 85° each. The 5th angle is 200°. Is it convex or non-convex?
   Solution: NON-CONVEX (concave)
   One angle (200°) is greater than 180° (reflex angle).
   Check: 4(85°) + 200° = 340° + 200° = 540° ✓ (matches pentagon sum: (5-2)×180° = 540°)
   Since one angle (200°) exceeds 180°, the pentagon is NON-CONVEX
8. Problem 8: You draw all diagonals in a polygon. Two diagonals pass outside the shape. Is it convex or non-convex?
   Solution: NON-CONVEX (concave)
   If ANY diagonal goes outside, the polygon is non-convex.
   Check: ✓
9. Problem 9: A polygon has 6 sides. When you extend any side, the polygon always stays on one side of the line. Is it convex or non-convex?
   Solution: CONVEX
   The extended side test shows all parts of the polygon stay on one side = convex.
   Check: ✓
10. Problem 10: Is a rectangle always convex?
    Solution: YES – always CONVEX
    All rectangles have four 90° angles (all < 180°), no indentations.
    Check: ✓

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Advanced Problems

11. Problem 11: A hexagon has interior angles: 120°, 120°, 120°, 120°, 120°, 120°. Is it convex or non-convex? Is it regular?
    Solution: CONVEX (possibly regular)
    All angles = 120° (< 180°) → convex
    Check sum: 6(120°) = 720° = (6-2)×180° ✓
    If all sides are also equal, it’s a regular hexagon.
    Answer: CONVEX
12. Problem 12: You have a polygon where one vertex « pokes inward. » Is it convex or non-convex?
    Solution: NON-CONVEX (concave)
    « Pokes inward » = indentation = creates a reflex angle > 180° at that vertex.
    Check: ✓
13. Problem 13: An octagon (8 sides) is regular. Is it convex or non-convex?
    Solution: CONVEX
    All regular polygons are convex. Regular octagon has all angles = 135° (< 180°).
    Formula: Interior angle = (n-2)×180°/n = (8-2)×180°/8 = 135°
    Check: ✓
14. Problem 14: True or False: A polygon with all sides equal must be convex.
    Solution: FALSE
    Having equal sides doesn’t guarantee convexity. You also need equal angles (making it regular).
    Example: You can create a concave polygon with equal sides but unequal angles.
    Check: ✓
15. Problem 15: A polygon has 10 vertices. You draw all possible diagonals, and every single diagonal stays inside. Is it convex or non-convex?
    Solution: CONVEX
    If ALL diagonals stay inside, the polygon is convex (no indentations for diagonals to escape).
    Check: ✓

What Mistakes Do Students Make With Convex and Non-Convex Polygons?

❌ Mistake #1: Thinking « convex » means « curved »

WRONG: « Convex means curved outward like a lens »
CORRECT: In polygons, convex means NO indentations and all angles < 180°. Polygons have straight sides, not curves.

❌ Mistake #2: Confusing reflex angles

WRONG: « An angle of 170° is reflex, so the polygon is concave »
CORRECT: Reflex angles are GREATER than 180°. An angle of 170° is still acute-obtuse (< 180°), so it doesn't make a polygon concave.

❌ Mistake #3: Assuming all quadrilaterals are convex

WRONG: « Quadrilaterals like squares and rectangles are convex, so all quadrilaterals must be convex »
CORRECT: Some quadrilaterals can be concave (like dart shapes or arrowheads).

❌ Mistake #4: Missing indentations

WRONG: Looking only at obvious « star » shapes and missing subtle indentations
CORRECT: Even a small dent or single reflex angle makes a polygon non-convex.

Why Do Convex and Non-Convex Polygons Matter in Math?

Understanding convex vs non-convex polygons is essential for:

• Geometry proofs: Properties differ between convex and concave polygons
• Area calculations: Different formulas may be needed
• Computer graphics: Rendering algorithms treat convex and concave shapes differently
• Architecture and design: Building shapes and structural considerations
• Quebec curriculum: Core Secondary 2-4 geometry requirement
• Advanced math: Foundation for trigonometry, calculus, and vector geometry

What Other Geometry Topics Should You Learn Next?

Now that you understand convex vs non-convex polygons, learn about:

• Polygon Types – triangle, quadrilateral, pentagon classification
• Interior Angles – sum formulas for polygons
• Regular Polygons – properties of equal-sided shapes
• Diagonals in Polygons – counting and properties

Frequently Asked Questions (FAQ)

Summary

Key Takeaways:

• Convex: All angles < 180°, no dents, all diagonals inside
• Non-convex (concave): At least one angle > 180°, has indentations, some diagonals outside
• All triangles are convex – no exceptions!
• All regular polygons are convex – equal sides + equal angles = convex
• Simple test: Look for dents or use the extended side test
• Key distinction: Reflex angles (> 180°) make polygons non-convex