Concave Polygons
Concave polygons are those polygons in which at least one interior angle is a reflex angle and it points inwards. They have a minimum of 4 sides and a few of the diagonals in a concave polygon may lie partly or fully outside it. All concave polygons are irregular because the interior angles are not equal. Let us learn more about concave polygons in this article.
1.  What is a Concave Polygon? 
2.  Concave and Convex Polygon 
3.  Concave Polygons in Real Life 
4.  FAQs on Concave Polygon 
What is a Concave Polygon?
A polygon that is not convex is a concave polygon. If any interior angle in a polygon measures more than 180°, then that is a concave polygon. A concave polygon looks like a polygon in which at least two sides seem to be pushed inwards. For example, a star is a concave polygon.
Concave Polygon Definition
A concave polygon is defined as a polygon in which one or more interior angles are more than 180°. If any of the diagonals of a polygon are formed partly or fully outside the polygon, then it is called a concave polygon. Observe the following figure which shows concave polygons.
Properties of Concave Polygon
A concave polygon has some distinct properties that help to distinguish it from other polygons.
 A concave polygon has at least one angle which is a reflex angle, that is, it is more than 180° and less than 360°.
 Some of the diagonals in a concave polygon may lie partly or fully outside it.
 There is at least one vertex in a concave polygon that seems to be pushed inwards to give that distinct 'cave' shape to the polygon.
 If a line segment is drawn across a concave polygon, it touches more than 2 sides.
 Regular polygons can never be concave.
 All concave polygons are irregular because the interior angles are of different measures.
Concave and Convex Polygon
A convex and concave polygon can be easily distinguished based on their properties. Observe the following figure and the table to see the difference between Convex and Concave polygons.
Convex Polygons  Concave Polygons 

A convex polygon has no interior angle that measures more than 180°  A concave polygon has at least one reflex angle (which measures more than 180°). 
A convex polygon can have 3 sides.  A concave polygon has at least 4 sides. 
When a line is drawn inside a convex shape from any one side, it touches only 2 sides.  When a line is drawn inside a concave shape from one side, it touches more than 2 sides. 
No diagonal in a convex polygon lies outside it.  Some of the diagonals in a concave polygon may lie partly or fully outside it. 
Concave Polygon Angles
A concave polygon has interior and exterior angles just like other polygons. The only property of concave polygons that should be remembered here is that one or more interior angles form a reflex angle which makes it look like a concave polygon. In other words, this reflex angle makes the vertex point inwards which gives it a peculiar shape. Let us read more about the sum of the exterior and interior angles of a concave polygon.
Sum of Exterior Angles of a Concave Polygon
The sum of the exterior angles of a concave polygon is 360°. This means just like all the other polygons, the exterior angles always add up to 360° for all concave polygons.
Sum of Interior Angles of a Concave Polygon
In order to find the sum of interior angles of a concave polygon, we use the same formula that is applied for other convex polygons. Sum of interior angles = (n  2) × 180°, where 'n' represents the number of sides of the polygon.
Concave Polygons in Real Life
Concave polygons have a noticeable shape in which at least one of the vertices is pointed inwards. We can notice concave polygons in our daytoday life. For example, a star, an arrowhead, a plus sign, and many more.
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Examples on Concave Polygon

Example 1:
Write true or false for the following statements.
a.) All the interior angles of a concave polygon are less than 180°.
b.) Concave shapes are those shapes in which at least one vertex points inwards.
Solution:
a.) False, at least one interior angle of a concave polygon is more than 180°.
b.) True, concave shapes are those shapes in which at least one vertex points inwards. This gives the distinct shape to a concave polygon.

Example 2:
Which of the following shapes represent a concave polygon?
a.) Star
b.) Arrow
c.) Rhombus
Solution:
a.) Star  Concave polygon
b.) Arrow  Concave polygon
c.) Rhombus  Convex polygon

Example 3: Fill in the blanks with the correct word.
a.) In a _____ polygon, some of the diagonals may lie partly or fully outside it.
b.) A ______ polygon can have 3 sides.
Solution:
a.) In a concave polygon, some of the diagonals may lie partly or fully outside it.
b.) A convex polygon can have 3 sides.
FAQs on Concave Polygon
What is a Concave Polygon in Geometry?
A concave polygon is defined as a polygon in which one or more interior angles is more than 180°. If any of the diagonals of a polygon are formed partly or fully outside the polygon, then it is called a concave polygon. There is at least one vertex in a concave polygon that seems to be pushed inwards to give that distinct 'cave' shape to the polygon.
What is the Sum of Exterior Angles of a Concave Polygon?
The sum of the exterior angles of a concave polygon is 360°. This means just like all the other polygons, the exterior angles always add up to 360° for all concave polygons.
What is the Sum of Interior Angles of a Concave Polygon?
In order to find the sum of interior angles of a concave polygon, we use the same formula that is applied for other convex polygons. Sum of interior angles = (n  2) × 180°, where 'n' represents the number of sides of the polygon.
What is an Example of a Concave Polygon?
We come across many reallife examples of concave polygons like a star, an arrowhead, and many more that have a peculiar shape and that satisfy all the characteristics of a concave polygon.
How to Find the Perimeter of a Concave Polygon?
The perimeter of a concave polygon can be calculated in the same way as the perimeter of other convex polygons is found. All the sides of the polygon are added to find the perimeter.
How to Find the Area of a Concave Polygon?
Since concave polygons are irregular in shape, there is no specific formula to find the area. A concave polygon can be divided into triangles or other smaller polygons as per the shape. Then, the area of these shapes can be calculated with the appropriate formulas and these areas can be added together to get the area of the concave polygon.
What is the Difference Between Concave and Convex Polygon?
The difference between concave and convex polygons can be listed as follows.
 A convex polygon has no interior angle that measures more than 180°, whereas, a concave polygon has at least one reflex angle which measures more than 180°.
 No diagonal in a convex polygon lies outside it. Some of the diagonals in a concave polygon may lie partly or fully outside it.
 A convex polygon can have 3 sides, whereas, a concave polygon has at least 4 sides.
Is a Star a Concave Polygon?
Yes, a star is a concave polygon because it satisfies all the properties of a concave polygon. It has more than 4 sides, it has angles that are more than 180° and the diagonals in a star lie outside the shape.
Is an Arrow a Concave Polygon?
Yes, an arrow is a concave polygon because it fulfills all the conditions of a concave polygon. It has interior angles that measure more than 180° and one diagonal in an arrow lies outside the shape.
Is a Triangle a Convex or a Concave Polygon?
A triangle is a convex polygon because it is a threesided figure. A concave polygon has at least 4 sides.
Is Square a Concave Polygon?
No, a square is not a concave polygon, it is a convex polygon because all the diagonals of a square lie inside it and there is no interior angle which is more than 180°.
Is a Concave Polygon Always Regular?
No, all concave polygons are irregular because one of their angles is a reflex angle. In other words, the interior angles of a concave polygon are not equal, therefore, all concave polygons are irregular.
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