A convex polygon is a polygon with straight sides in which every interior angle is less than (
Types
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.
Interior Angles
The angles inside a polygon are known as its interior angles. A polygon has as many internal angles as it does sides.
Exterior Angles
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 VS Concave Polygon
Some of the key differences between convex and concave polygon are listed below:
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. |
Formulas
Let's look at some convex polygon formulas:
Regular 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), . . . , (xn, yn), then the formula for finding its area is
Area = ½ |(x1y2 - x2y1) + (x2y3 - x3y2) + . . . + (xny1 - x1yn)|
Regular 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 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)
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.
Example: A pentagon has 5 sides. This means n = 5
So, the result is,
Each exterior angle is 360°/5 =72°
Properties
Some of the important properties of convex polygons are:
- 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°.