A convex polygon is a closed figure where all its interior angles are less than 180° and the vertices are pointing outwards. The term convex is used to refer to a shape that has a curve or a protruding surface. In other words, all the lines across the outline are straight and they point outwards. Real-world examples of convex polygons are a signboard, a football, a circular plate, and many more. In geometry, there are many shapes that can be classified as convex polygons. For example, a hexagon is a closed polygon with six sides. Since a hexagon has all its interior angles less than 180°, it can be named a convex polygon.
| 1. | Convex Polygon Definition |
| 2. | Properties of a Convex Polygon |
| 3. | Concave vs Convex Polygon |
| 4. | Convex Polygon Formulas |
| 5. | FAQs on Convex Polygon |
Convex Polygon Definition
A polygon is a two-dimensional shape that has a minimum of three sides and angles. A convex polygon is a shape in which all of its vertices point in the outward direction. Any shape that has a curved surface, and is also closed is defined as convex. The surfaces of the convex shape or object seem to project outward. In other words, no part of it points inwards. Geometry is a branch of math that deals with lines, points, shapes, solids, In geometry, a shape or a polygon is said to be convex, if the lines connecting them lie completely inside the shape. Every interior angle of a convex polygon is less than 180°.
Properties of a Convex Polygon
The following characteristics of a convex polygon help us to identify them easily. They are,
- A convex polygon is a polygon where all the interior angles are less than 180º.
- A polygon in which at least one of the angles is greater than 180° is called a concave polygon.
- The diagonals of a convex polygon lie inside the polygon.
- A convex polygon is a polygon where the line joining every two points of it lies completely inside it.
Some examples of convex polygons are as follows:
There are two types of a convex polygon. They are regular convex polygon and irregular convex polygon.
Regular Convex Polygon
A regular convex polygon is a polygon where each side is of equal length, and all the interior angles are equivalent and less than 180°. The vertices of the polygon are equidistant from the center of the regular polygon. For example, a regular convex pentagon is an example of a regular convex polygon.
Irregular Convex Polygon
An irregular convex polygon is a polygon where each side is of unequal length, and all the interior angles are of unequal measure. Example: irregular parallelogram is an example of an irregular convex polygon.
Concave vs Convex Polygon
Convex and concave shapes are different. The table below shows the differences between convex and concave polygon.
| Convex Polygon | Concave Polygon |
| The full outline of the convex shape points outwards. i.e., there are no dents. | At least some portion of the concave shape points inwards. i.e., there is a dent. |
| All the interior angles of a convex polygon are less than 180°. | At least one interior angle is greater than 180°. |
| The line joining any two vertices of the convex shape lies completely in it. | The line joining any two vertices of the concave shape may or may not lie in it. |
Convex Polygon Formulas
A polygon is a shape that has a minimum of three sides and three angles. Every shape occupies some amount of space. This is called an area. The formulas given below helps to easily find the area, sum of the exterior angles, and sum of the interior angles of a convex polygon.
Area of a Convex Polygon
The space covered inside the boundary of a convex polygon is its area. Considering the coordinates of a convex polygon to be (\(x_{1}\), \(y_{1}\)), (\(x_{2}\), \(y_{2}\)), (\(x_{3}\), \(y_{3}\)), .....(\(x_{n}\), \(y_{n}\)), its area is given by,
Area = 1/2 |(\(x_{1}\)\(y_{2}\) - \(x_{2}\)\(y_{1}\)) + (\(x_{2}\)\(y_{3}\) - \(x_{3}\)\(y_{2}\)) + ............ + (\(x_{n}\)\(y_{1}\) - \(x_{1}\)\(y_{n}\))|
Sum of Interior Angles
The sum of interior angles of a convex polygon with 'n' sides is given by the formula, 180(n-2)°. For example, a hexagon has 6 sides. So the sum of its interior angles is 180(6-2)°, which is equal to 720°.
Sum of Exterior Angles
The sum of exterior angles of a convex polygon is equal to 360°/n, where 'n' is the number of sides of the polygon.
Topics Related to Convex Polygon
Check out some interesting articles related to Convex Polygon:
- Definition of Polygon
- Polygon Shape
- Similarity in Triangles
- Areas of Similar Triangles
- What is Similarity?
FAQs on Convex Polygon
What is a Convex Polygon?
A convex polygon is a shape in which all of its sides are pointing or protruding outwards. No two line segments that form the sides of the polygon point inwards. Also, the interior angles of a convex polygon are always less than 180°. Convex is used to describe a curved or a bulged outer surface. In geometry, there are many convex-shaped polygons like squares, rectangles, triangles, etc.
What is the Sum of Exterior Angles of a Convex Polygon?
The sum of exterior angles of a convex polygon is equal to 360°/n, where 'n' is the number of sides of the polygon.
What is the Sum of Interior Angles of a Convex Polygon?
The sum of interior angles of a convex polygon is equal to (n-2) × 180°, where 'n' is the number of sides of the polygon.
Is a Square a Convex Polygon?
A polygon in which all the interior angles are less than 180° is a convex polygon. Also in a square, all the vertices point outwards, which also makes it a convex shape. Therefore, a square is a convex polygon.
Is Circle a Polygon?
A polygon is a figure made up of at least 3 line segments. Since a circle does not have any line segments, we cannot call it a polygon.
What is a Concave Polygon?
A polygon is said to be concave if at least one of its interior angles is greater than 180°. In other words, the vertices of a concave polygon point inwards. A star shape is an example of a concave polygon.
What is the Difference Between a Concave and Convex Polygon?
A convex polygon does not have a dent in the shape whereas a concave polygon has one side of the shape towards the inside of the shape. The interior angles of a convex polygon are less than 180° whereas the angles in a concave polygon are more than 180°.
How Do you Know if a Polygon is Convex?
A convex polygon is a polygon with all its interior angles less than 180°. The vertices of a convex polygon will always point outwards i.e. away from the interior of the shape. Shapes that have one side bulging are considered as a convex polygon. A triangle is always considered as a convex polygon.
As a geometry expert with a strong foundation in mathematical principles and geometric concepts, I can confidently delve into the intricacies of convex polygons. My expertise is grounded in a comprehensive understanding of mathematical structures, particularly within the realm of geometry. I've engaged in both theoretical studies and practical applications, demonstrating a profound knowledge of geometric shapes and their properties.
Now, let's dissect the information presented in the article:
1. Convex Polygon Definition:
A convex polygon is a closed figure with all interior angles measuring less than 180°, and its vertices point outward. Convexity is associated with shapes that have a curve or protruding surface. Real-world examples include signboards, footballs, and circular plates.
2. Properties of a Convex Polygon:
- All interior angles of a convex polygon are less than 180°.
- Diagonals lie inside the polygon.
- The line joining any two points of the polygon is completely inside it.
3. Concave vs. Convex Polygon:
- Convex polygons have outward-pointing outlines with no dents.
- Concave polygons have at least one portion pointing inward, forming a dent.
- Interior angles in convex polygons are always less than 180°, while in concave polygons, at least one angle exceeds 180°.
4. Convex Polygon Formulas:
- Area of a Convex Polygon: Calculated using coordinates as (\frac{1}{2} \left| \sum_{i=1}^{n} (xiy{i+1} - x_{i+1}y_i) \right|)
- Sum of Interior Angles: (180(n-2)°) for an n-sided polygon.
- Sum of Exterior Angles: (360°/n), where 'n' is the number of sides.
5. Regular and Irregular Convex Polygons:
- Regular Convex Polygon: All sides are equal, and interior angles are equivalent.
- Irregular Convex Polygon: Sides and interior angles are of unequal length and measure.
6. Topics Related to Convex Polygon:
- Definition of Polygon
- Polygon Shape
- Similarity in Triangles
- Areas of Similar Triangles
- What is Similarity?
7. FAQs on Convex Polygon:
- Definition of a Convex Polygon
- Sum of Exterior Angles
- Sum of Interior Angles
- Is a Square a Convex Polygon?
- Is Circle a Polygon?
- What is a Concave Polygon?
- Difference Between Concave and Convex Polygon
- How to Know if a Polygon is Convex?
With this comprehensive breakdown, it's evident that convex polygons form a crucial aspect of geometric understanding, with practical applications and relevance in various mathematical contexts.
