What Is a Convex Polygon Definition Formula and Properties
What is Convex Polygon?
We will see the definition of a convex polygon, but let us begin with a reminder of what a polygon is? It is any two-dimensional shape with straight lines and angles. So, Triangle is a polygon and so is the square. Convex polygon examples are in plenty, most of them are even not discussed regularly. Their monikers are generally based on the number of sides of the concerned 2D shape in question. Convex polygon definition is quite simple and easy to understand. A convex polygon is 2D shaped with all the interior angles less than 180-degree. A prime example of a convex polygon would be a triangle. The vertices of a convex polygon bulge away from the interior angle. It is the most important factor, which makes spotting a convex polygon definition easier.
Here, we will discuss specifically convex polygons. We will try to understand how you define convex polygon?
We also discuss some of the properties of convex polygons that separate from the rest of the shapes? Also, how to calculate the area of a polygon? So let us jump right into it.
Properties of Convex Polygon
There are three crucial properties of convex polygon which are mentioned below.
In the case of a convex polygon, the sum of the internal angles is represented in the form of (n-2) 180*. Here, ‘n’ is the number of sides of a polygon.
All the interiors angles of a polygon are less than 180*
Concave polygon is just the opposite of convex polygon with at least one side more than 180*.
Types of Polygons
It is very important to have an answer to “what is a convex polygon”. This makes way for the type of polygons- regular and irregular.
A regular polygon is one which has all the sides of equal length, while in case of irregular polygons the length of the sides is not the same. Triangle in a convex polygon, and it has the special property of being both regular and irregular.
Case in point, a rectangle or a square cannot be both regular and irregular polygon at the same time.
Examples of regular polygon- square, and equilateral triangle.
Example of irregular polygon- rectangle. And scalene triangle.
Can a Pentagon be Both Convex and Concave?
Pentagon is any five-sided 2D shape which can be drawn on an XY plane. In case of a pentagon, the sum of interior angles can show up to 540*.
In an ideal scenario, there are both concave and convex pentagons. Most students in the initial stages of learning form a false notion that the pentagon is rigidly but convex.
Theorems
If we have a sum of all the interior angles of a regular polygon, then calculating the value of the individual interior angle is quite simple. You just need to divide the sum of all angles by the number of sides.
In the case of a regular polygon with n sides, the sum of exterior angles is always equal to 360*. Thus, if we are told to find the value of an exterior angle, we just need to divide the sum of the exterior by the number of sides.
How can you calculate the area of a convex Polygon?
A=½ |(A1 B2 - A2 B1)+(A2 B3 - B3 A2)+……..+(An B1-- Bn A1)|
Using the above equation you can easily find the area of a regular convex polygon with vertices (A1, B1) , (A2, B2) ,...... (An. Bn).
Solved Examples
1. Find the Area of a Regular Polygon with Three Sides Whose Vertices are: (7, 9), (5, 2) and (-4, 5).
Here , (A1, B1)= (7,9), and
(A2, B2)= (5, 2), and
(A3, B3)= (-4, 5).
The formula to find the area of a convex polygon is
A=½ |(A1 B2 - A2 B1)+(A2 B3 - B2 A3)+……..+(A3B1-- B3 A1)|
A = ½ | (14-45) + (25+8 ) + (-36-35)|
A = ½ |-73|
A = 73/2
Therefore, the area of the convex polygon is 73/2.
2. Find the Area of a Regular Polygon with Three Sides whose Vertices are: (10, 7), (4, 2) and (-2, 4)
Here , (A1, B1)= (10,7), and
(A2, B2)= (4, 2), and
(A3, B3)= (-2, 4).
The formula to find the area of a convex polygon is
A=½ |(A1 B2 - A2 B1)+(A2 B3 - B2 A3) + (A3B1-- B3 A1)|
A = ½ | (20-28) + (16+4 ) + (-14-40)|
A = ½ |66|
A = 66/2
Therefore, the area of the convex polygon is 73/2.
Did you know
Most of the irregular polygons are not actively taught and there a lot we do not know about such shapes.
The term polygon has its origin from Greek word ‘poly’ meaning many and ‘gonia’ meaning angles.
FAQs on Convex Polygon Meaning Properties and Examples
1. What is a convex polygon?
A convex polygon is a polygon in which all interior angles are less than 180° and all diagonals lie inside the shape. This means the polygon has no inward dents or indentations.
- Every interior angle is < 180°.
- Any line segment joining two points inside the polygon stays inside it.
- Examples include a triangle, square, and regular pentagon.
2. How do you identify if a polygon is convex or concave?
A polygon is convex if all its interior angles are less than 180°, and concave if at least one interior angle is greater than 180°. To check:
- Measure each interior angle.
- If any angle is > 180°, the polygon is concave.
- Alternatively, draw diagonals—if any diagonal lies outside the shape, it is concave.
3. What is the formula for the sum of interior angles of a convex polygon?
The sum of interior angles of a convex polygon with n sides is (n − 2) × 180°. For example:
- For a pentagon (n = 5): (5 − 2) × 180° = 540°.
- For a hexagon (n = 6): (6 − 2) × 180° = 720°.
4. What is each interior angle of a regular convex polygon?
Each interior angle of a regular convex polygon is given by [(n − 2) × 180°] / n. Since all sides and angles are equal in a regular polygon:
- For a regular hexagon (n = 6): (4 × 180°) / 6 = 120°.
- For a square (n = 4): (2 × 180°) / 4 = 90°.
5. How many diagonals does a convex polygon have?
A convex polygon with n sides has n(n − 3) / 2 diagonals. This formula counts all line segments joining non-adjacent vertices.
- For a pentagon (n = 5): 5(5 − 3)/2 = 5 diagonals.
- For a hexagon (n = 6): 6(6 − 3)/2 = 9 diagonals.
6. Can a triangle be a convex polygon?
Yes, every triangle is a convex polygon because all its interior angles are less than 180°. In fact:
- The sum of angles in a triangle is 180°.
- No triangle can have an interior angle greater than 180°.
- All diagonals (none in this case) would lie inside the shape.
7. What is the difference between a convex polygon and a regular polygon?
A convex polygon has all interior angles less than 180°, while a regular polygon has all sides and angles equal. Key differences:
- A convex polygon may have unequal sides and angles.
- A regular polygon is always convex (if simple).
- Example: A rectangle is convex but not regular (unless it is a square).
8. What are the properties of a convex polygon?
The main properties of a convex polygon relate to its angles and diagonals. These include:
- All interior angles are < 180°.
- All diagonals lie inside the polygon.
- The sum of interior angles is (n − 2) × 180°.
- It has no indentations or inward curves.
9. How do you calculate the exterior angle of a convex polygon?
Each exterior angle of a regular convex polygon is 360° / n. Also, the sum of all exterior angles of any convex polygon is always 360°.
- For a regular pentagon: 360° / 5 = 72°.
- For a regular octagon: 360° / 8 = 45°.
10. What are some real-life examples of convex polygons?
Many everyday shapes are examples of convex polygons, meaning they have no inward dents. Common examples include:
- A square tile (4-sided convex polygon).
- A triangular road sign (3-sided convex polygon).
- A hexagonal honeycomb cell (6-sided convex polygon).
