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Convex Polygon
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. Realworld 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 twodimensional 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(n2)°. For example, a hexagon has 6 sides. So the sum of its interior angles is 180(62)°, 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
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Convex Polygon Examples

Example 1:What is the measure of an interior angle of a regular convex polygon like a pentagon?
Solution:
The number of sides of a pentagon is n =5. We know that the sum of all interior angles of a polygon of n sides is 180(n2)° degrees. Hence, the sum of the interior angles of the pentagon is:
180 × (52)° =180 × (3)°
= 540°
Since the given pentagon is regular, all 5 interior angles measure the same. Therefore, the measure of each interior angle is 540° / 5 = 108°
Therefore, the required angle = 108°.

Example 2:Find the convex polygon from the following figures.
Solution:
Among the given shapes, (a), (d), and (f) have all their vertices pointing outwards. Also in (a), (d), and (f), all the interior angles measure less than 180°. Therefore, these are the only convex polygon among the given shapes. Therefore, a), d) and f) are convex polygons.

Example 3: Find the area of a regular convex polygon whose vertices are (6,8), (4.1), and (3, 6).
Solution:
Area of a regular polygon is given by the formula,
Area = 1/2 (\(x_{1}\)\(y_{2}\)  \(x_{2}\)\(y_{1}\)) + (\(x_{2}\)\(y_{3}\)  \(x_{3}\)\(y_{2}\)) + ............ + (\(x_{n}\)\(y_{1}\)  \(x_{1}\)\(y_{n}\))
The vertices are, (\(x_{1}\), \(y_{1}\)) = (6,8) , (\(x_{2}\), \(y_{2}\)) = (4,1) , (\(x_{3}\), \(y_{3}\)) = (3,6)
Substituting the values we get,
Area = 1/2 (6  32) +(24  (3)) + (24  36)
= 1/2 26 + 27  60
= 1/2 59
= 1/2 59
= 59/2 square units
Therefore, the area of the convex polygon is 59/2 square units.
Practice Questions on Convex Polygon
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 convexshaped 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 (n2) × 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.
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