Convex Polygon: A Complete and Easy Guide

Polygons are everywhere in our daily lives. Shapes like triangles, rectangles, and regular pentagons are simple examples of polygons. But when we talk about convex polygons, many students get confused. Do not worry! This guide will explain convex polygons in the easiest way possible.

Also Read : Polygon bike

 What Is a Convex Polygon?

A convex polygon is a polygon where all angles point outward and every interior angle is less than 180°. If you draw a line between any two points inside the shape, the line will always stay inside the polygon.

Simple meaning: A polygon that doesn’t bend inward anywhere.

Examples of convex polygons:

• Triangle
• Square
• Regular pentagon
• Regular hexagon

These shapes look smooth with no inward corners.

 How to Identify a Convex Polygon?

You can check a polygon easily using these rules:

ConditionMeaning All interior angles < 180° No angle bends inward. No dents or caves Shape looks outward. The line between any two points stays inside. No part sticks out of the shape.

If any angle goes inside like a “cave,” then it is not convex.

 Difference Between Convex and Concave Polygons

Feature Convex Polygon Concave Polygon Angles: All < 180°At least one > 180° Shape 180° AtOutward and smooth Has an inward dent line test always inside Line can go outside. ExampleTriangle Star-shaped figure

A concave polygon looks like it has a bite taken out of it!

 Types of Convex Polygons

Convex polygons can be divided into two kinds:

 Regular Convex Polygon

All sides and angles are equal.
Examples: Square, regular hexagon

 Irregular Convex Polygon

Sides and angles are not equal.
Examples: Some quadrilaterals, uneven pentagons

Both still point outward with no inward corners.

 Properties of Convex Polygons (Easy Rules)

Here are the important properties:

• Interior angles are less than 180°.
• Exterior angles add up to 360°.
• Shape is always simple (not crossed)
• Diagonal stays inside the shape
• More sides → more interior angle sum

These rules help in geometry problem solving.

 Angle Sum of a Convex Polygon

To find the sum of the interior angles:

Sum=(n−2)×180∘\text{Sum} = (n – 2) \times 180^\circSum=(n−2)×180∘

Where n = number of sides

Example:
Hexagon → n = 6

(6−2)×180∘=4×180∘=720∘(6 – 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ(6−2)×180∘=4×180∘=720∘

So, the sum of angles in a convex hexagon = 720°.

 Exterior Angles of a Convex Polygon

The exterior angle sum is always:

360∘360^\circ360∘

Even if it has 10 or 100 sides, the exterior angles will always add up to 360°.

 Area of a Convex Polygon

Area depends on the type of polygon:

• Triangles: ½ × base × height
• Rectangles: length × width
• Regular polygons: special formulas using side or radius

For irregular convex polygons, divide into smaller shapes to find area.

 Real-life Examples of Convex Polygons

We see many convex shapes around us:

• Tiles on the floor
• Road signs like STOP (octagon)
• Honeycomb cells (hexagon)
• Pieces of chocolate bars
• Windows and picture frames

Convex shapes are super useful in design, architecture, and construction.

 Quick Summary Table

Topic, meaning, definition All angles < 180°, no inward dentDiagonals Stay inside. Angle sum (n − 2) × 180° Exterior angle sum: 360° Real uses Buildings, signs, tiles

 FAQs

 Q1: Can a triangle be a convex polygon?

➡ Yes! All triangles are convex.

 Q2: What is the smallest convex polygon?

➡ A triangle (3 sides) is the smallest possible polygon.

Q3: How do we know a polygon is concave?

➡ If even one interior angle is greater than 180°, it becomes concave.

Q4: Are all regular polygons convex?

➡ Yes! Regular shapes always point outward.

Q5: Can convex polygons be irregular?

➡ Of course! As long as no angle bends inward, it is still convex.

Final Words

Convex polygons are simple shapes with outward-facing angles. They are the building blocks of many structures we use every day. If a shape has no inward dents, then you are looking at a convex polygon.