Convex Polygon

Convex polygons are special shapes with straight sides that don’t fold inward. All the lines connecting the corners of a convex polygon stay inside the shape. The corners of a convex polygon always point outward. A regular polygon, where all sides and angles are equal, is always convex. If a closed shape has a curved surface, it’s not a convex polygon. In geometry, a polygon is a flat, two-dimensional shape with straight sides and angles. There are two main types of polygons: convex and concave, based on the angles inside. Let’s explore more about convex polygons, including their properties, types, formulas, and examples.

Convex Polygon Definition

A closed figure known as a convex polygon is one in which all of the inner angles are less than 180°. As a result, the polygon‘s vertices will all face away from the shape’s interior, or outward. No sides are inwardly pointed. A triangle is thought to be a significant convex polygon. The quadrilateral, pentagon, hexagon, parallelogram, and other convex polygonal forms are examples.

Convex Polygon Examples

Convex polygons are frequently used in both architecture and daily life. Examples are stop signs, which have eight sides and an outward curve, and many modern buildings’ windows, which also have convex surfaces. Crosswalks and pedestrian islands on roads are commonly marked with convex polygonal forms. Dinner plates and tabletops, which have a smooth and outwardly bulging shape, can also serve as examples of convex polygons.

Types of Convex Polygons

Two categories can be used to categorize convex polygons. These are what they are:

• Regular Convex Polygon
• Irregular Convex Polygon

Regular Convex Polygon

A regular convex polygon is one with all sides equal and all interior angles equal. Regular polygons can only be convex polygons. Every concave polygon is irregular. As a result, while all regular polygons are convex, not all convex polygons are regular.

Irregular Convex Polygon

The sides and angles of irregular polygons vary in length. If all of the irregular polygon’s angles are fewer than 180 degrees, it is an irregular convex polygon. An example of an irregular convex polygon is the scalene triangle.

Properties of Convex Polygons

• A convex polygon’s diagonals are located within the polygon.
• A convex polygon is one in which the line connecting every two points is entirely within it.
• A convex polygon has all of its internal angles less than 180°.
• A concave polygon is one that has at least one angle that is larger than 180°.

Interior Angles of Convex Polygon

The angles inside a polygon are known as its interior angles. A polygon has as many internal angles as it does sides.

Exterior Angles of Convex Polygon

The exterior angles of a polygon are the angles obtained by extending a polygon’s sides. The sum of a polygon’s outside angles is 360°.

Convex Polygon and Concave Polygon

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

The 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° x (n-2)

For example:

A pentagon has five sides. Which means n = 5

So, the result,

The total of its interior angles is 180° x (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°.

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°

People Also Read:

• Polygon Formula – Definition, Symbol, Examples
• Polygon – Shape, Formula, Types, and Examples
• Types of Polygons: Classification and Table
• Area of Polygon

FAQs on Convex Polygons

Q1: What is a Convex Polygon?

Answer:

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.

Q2: How can we know whether a polygon is concave or convex?

Answer:

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.

Q3: What polygonal shapes fall under which categories?

Answer:

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

Q4: Which type of polygon is a triangle—convex or concave?

Answer:

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.

Q5: What are some examples of convex polygons in the real world?

Answer:

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.