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. Some of the common 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
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°.
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
Some of the key differences between convex and concave polygon are listed below:
Convex 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. |
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), . . . , (xn, yn), then the formula for finding its area is
Area = ½ |(x1y2 - x2y1) + (x2y3 - x3y2) + . . . + (xny1 - x1yn)|
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,
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.
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.
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