Convex Polygon with Examples and FAQs

Convex Polygon Definition

A Polygon is a closed two-dimensional figure which has more than two sides, vertices and interior angles. The polygons which have all the vertices pointing outwards and each of its interior angles measuring less than 180° are convex polygons.

The images below are some examples of such polygons:

Note:

Concave Polygon

The polygons which have vertices which are a combination of both inwards and outwards and at least one angle measuring greater than 180° are concave polygons.

The images below are some examples of concave polygons.

Properties

The properties of convex polygons are:

• All interior angles measure less than 180°.
• All diagonals lie inside the polygon itself.
• The line joining any two points on the polygon lies completely inside the polygon. 

Sum of Angles of a Convex Polygon:

Interior angles

The sum of the interior angles of a convex polygon with ‘n’ number of sides can be found out by the formula:

[180\times (n-2)]^\circ

If the polygon is regular it will have all sides and interior angles of equal measure.

For example, we know for a triangle sum of interior angles is 180°, let us verify this using the formula above:

For a triangle the value of n = 3,

Sum of interior angles =[180 \times(n-2)]^{\circ}=[180 \times(3-2)]^{\circ}=[180 \times 1]^{\circ}=180^{\circ} .

Exterior angles

The sum of all exterior angles of a convex polygon is 360°, irrespective of the number of sides of the polygon.

For a regular polygon i.e., when the measure of all sides is equivalent, each exterior angle is equal to \frac{360^{\circ}}{n}, where ‘n’ is the number of sides.

Number of diagonals

The number of diagonals for a polygon having ‘n’ sides is given by the formula:

\frac{n(n-3)}{2}

Examples

Example 1: Can you determine which polygons are convex in nature from the options given.

Solution: (A) and (C) are convex polygons, whereas (B) and (D) are concave polygons.

Example 2: If a polygon that is convex has 10 sides, then what is the sum of its interior angles? 

Solution:

The sum of interior angles of a convex polygon with ‘n’ sides is given by the formula:

[180\times (n-2)]^\circ.

The given polygon has 10 sides. Hence, the total sum of all its angles

= [180 × (10 – 2)]°

= [180 × 8]°

= 1440°

Hence, the sum of all the interior angles of the polygon is 1440°.