Table of Content

• Convex Polygon Definition
• Convex Polygon Examples
• Types of Convex Polygons
• Regular Convex Polygon
• Irregular Convex Polygon
• Properties of Convex Polygons
• Convex Polygon and Concave Polygon
• Convex Polygon Formulas
• Regular Convex Polygon Area
• Regular Convex Polygon Perimeter
• Irregular Convex Polygon Perimeter
• The sum of Interior Angles
• Sum of Exterior Angles
• FAQs

Convex Polygon	Concave Polygon
Each interior angle of a convex polygon is less than 180 degrees.	At least one interior angle of a concave polygon exceeds 180 degrees.
It contains the line connecting any two vertices of the convex form.	It may or may not contain the line connecting any two vertices of the concave form.
The convex shape’s whole outline points outward. That is, there are no dents.	At least some of the concave curve is pointing inward. That is, there is a dent.
Both regular and irregular convex polygons exist.	Regular concave polygons never exist.

Convex Polygon Formulas

Let’s look at some convex polygon formulas:

Regular Convex Polygon Area

The formula for calculating the area of a regular convex polygon is as follows:

If the convex polygon includes vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), . . . , (x_n, y_n), then the formula for finding its area is

Area = ½ |(x₁y₂ – x₂y₁) + (x₂y₃ – x₃y₂) + . . . + (x_ny₁ – x₁y_n)|

Regular Convex Polygon Perimeter

The perimeter of a closed figure is the entire distance between its exterior sides. It is the entire length of a polygon’s sides. A regular polygon has sides that are equal on all sides. As a result, by repeated addition, we may get the perimeter of a regular polygon.

The perimeter of a regular convex polygon may be calculated using the following formula:

Perimeter = n x 8

Where n is the number of sides and 8 is the length of sides.

Irregular Convex Polygon Perimeter

It is the entire distance circumscribed by a polygon. It can be found simply by adding all of the polygon’s sides together.

The perimeter of an irregular convex polygon may be calculated using the following formula:

Perimeter = The total of all sides

Sum of Interior Angles

The sum of a convex polygon’s interior angles with ‘n’ sides. The sum of Interior Angles may be calculated using the following formula:

180° × (n-2)

For example: A pentagon has five sides. Which means n = 5

So, the result,

The total of its interior angles is 180° × (5-2) = 540°.

Sum of Exterior Angles

The sum of a convex polygon’s exterior angles is equal to 360°/n, where ‘n’ is the number of sides of the polygon.

For example: A pentagon has 5 sides. This means n = 5

So, the result is,

Each exterior angle is 360°/5 =72°.

Related Read,

• Polygon
• Types of Polygons
• Area of Polygon

Solved Examples of Convex Polygons

Example 1: Calculate the area of the polygon with vertices (2, 5), (6, 4), and (7, 3).

Solution:

Given: The vertices are (2, -5), (6, 4), and (7, 3)

Here , (x₁, y₁) = (2, -5)

(x₂, y₂) = (6, 4)

(x₃, y₃) = (7, 3)

The formula for calculating a convex polygon’s area is

Area = ½ | (x₁y₂ – x₂y₁) + (x₂y₃ – x₃y₂) + (x₃y₁ – x₁y₃) |

⇒ Area = ½ | (8+30) + (18 -28) +(-35-6) |

⇒ Area = ½ | -13|

⇒ Area = ½ |13|

Thus, Area = 13/2 Square Units

Example 2: Find the perimeter of a regular decagon whose side length is 30 m.

Solution:

A decagon is a polygon with 10 sides.

• s = 30
• n = 10

Perimeter = 10 × 30

⇒ Perimeter = 300m

Example 3: Find the Exterior angles of a pentagon are 2x°, 5x°, x°, 4x°, and 3x°. What is x?

Solution:

The sum of any convex polygon’s exterior angles is always 360°.

2x°+ 5x° + x° + 4x° + 3x° = 360°

⇒ 15x = 360°

⇒ x = 360°/15

⇒ x = 24.

Thus, the angles are:

2x = 48°, 5x = 120°, x = 24°, 4x = 96° and 3x = 72°

Example 4: Find the Interior angles of a pentagon are 2x°, 5x°, x°, 4x°, and 3x°. What is x?

Solution:

We know that a pentagon has 5 sides. So the interior angle is (5-2) x 180° = 540°
2x°+ 5x° + x° + 4x° + 3x° = 540°

⇒ 15x° = 540°

⇒ x = 540/15

⇒ x = 36°

Practice Problems on Convex Polygon

Problem 1: Calculate the sum of the interior angles of a convex hexagon.

Problem 2: Determine the number of sides in a convex polygon if the sum of its interior angles is 1080°.

Problem 3: Given a convex pentagon with one angle measuring 120°, find the measure of each of its other angles.

Problem 4: If a convex octagon has exterior angles measuring 45° each, what is the sum of its exterior angles?

Problem 5: Find the measure of each interior angle in a convex heptagon.

Problem 6: Calculate the number of diagonals that can be drawn from a single vertex in a convex decagon.

Problem 7: Determine the measure of an exterior angle of a regular convex nonagon.

FAQs on Convex Polygons

What is a Convex Polygon?

Any form with inner angles that are all less than 180 degrees is considered a convex polygon. A convex polygon won’t have any vertices that point inward, and all diagonal connecting lines will be contained inside the form.

How can we know whether a polygon is concave or convex?

If the internal angles of the polygon are less than 180 degrees, it is considered to be convex. However, the polygon is concave if any internal angle is greater than 180 degrees.

What polygonal shapes fall under which categories?

A polygon can be classified as convex, concave, regular, irregular, simple, or complicated depending on its shape.

Which type of polygon is a triangle—convex or concave?

Any polygon that has internal angles that are all smaller than 180 degrees is said to be convex. The polygon is concave if one or more of the inner angles exceed 180 degrees. Three sides make up a triangle, a geometric form. The sum of a triangle’s interior angles is 180 degrees. Hence, a triangle is a convex polygon.

What are some examples of convex polygons in the real world?

Convex polygons are ubiquitous in our surrounds and daily lives. Stop signs on roadways, the hexagons and pentagons on a football, a coin, etc. are a few examples from everyday life.