Concave Polygon

Concave Polygon is a type of polygon with at least one interior angle that is larger than 180°. In other words, a concave polygon has at least one “dent” or indentation in its boundary. In this article, we will learn about Concave Polygon in detail including their properties as well as formulas related to it’s interior as well as exterior angles.

What is a Concave Polygon?

Concave polygons can also be known as non-convex polygons or reentrant polygons, in which polygons have any one interior angle that measures greater than 180°.

Whereas, Its vertices either go inwards or outwards at a time. In other words, we can say it’s an extremely opposite polygon to a convex polygon. Even though, the diagonals of this polygon lie completely or partially outside the polygon.

Concave Polygon Definition

A Concave polygon is a polygon in which at least one interior angle measures more than 180°.

For example, stars can be represented as concave polygons because it appears as if two lines are pushed inwards while another line is pushed outwards. (In this example, the inward-pointing vertices refer to interior angles that exceed 180°).

Properties of Concave Polygon

A concave polygon can be distinguished from other polygons by a few unique characteristics or properties;

• In this polygon, at least one angle must be the reflex angle, that is more than 180° and less than 360°.
• In the shape of the polygon, there must be at least one vertex that seems to be pushed inwards.
• A line segment touches more than two sides of a concave polygon when it is drawn across it.
• These polygons can never be a regular polygon.
• Due to the various measurements of the inner angles why they are said to be irregular polygons.

Types of Concave Polygon

Concave polygons can be categorised as:

• Regular Concave Polygon
• Irregular Concave Polygon

Regular Concave Polygon

A polygon is said to be regular when its side lengths and interior angles both are equal. Even, any polygon must have at least one interior angle that will be more than 180 degrees then only it’s said to be a concave polygon. Moreover, the total of the interior angles of a polygon is equal to (n – 2) x 180, where “n” is the number of sides.

Irregular Concave Polygon

The irregular polygon’s sides have various measurements of each interior angle that are not even congruent because the interior angles of concave polygons and their irregularity are commonly observed. Whereas, The measurements of the interior angles as well as the lengths of the sides of irregular concave polygons can fluctuate and give rise to a shape that lacks regular symmetry or pattern.

Angles in Concave Polygon

Like any 2d geometric object, polygons can have interior as well as exterior angles.

• Interior Angle: An interior angle is an angle formed between two adjacent sides of a polygon, measured inside the polygon.
• Exterior Angle: An exterior angle of a polygon is an angle formed between one side of the polygon and the extension of an adjacent side, measured outside the polygon.

Sum of Exterior Angles of a Concave Polygon

In a polygon with n sides, each exterior angle corresponds to one interior angle. The sum of the exterior angles of any polygon, regardless of the number of sides, is always 360°.

Thus, for concave polygon sum of all exterior angle is 360°.

Sum of Interior Angles of a Concave Polygon

The sum of the interior angles of a concave polygon can be found by using the same formula as: (n-2) × 180°, Here “n” is the number of sides.

Example: Find the sum of the interior angles of a concave polygon with 7 sides.

Solution:

Given: n = 7

Sum of the Interior Angles = (n-2) × 180°

⇒ Sum of the Interior Angles = (7-2) × 180°

⇒ Sum of the Interior Angles = 5 × 180° = 900°

Concave Polygon Formulas

For any polygon, we can calculate

• Perimeter
• Area

Let’s discuss the formula for perimeter and area od concave polygons.

Perimeter of a Concave Polygon

The perimeter of a concave polygon are determined by the whole distance covered by their boundaries. Whereas, The length of each side of a given figure can be added together to determine its perimeter i.e.,

Perimeter of Concave Polygon = Sum of all sides given in a figure

Area of a Concave Polygon

The area of a concave polygon cannot be easily calculated but It is possible for each side and each interior angle to have a decided length. Thus, we must divide the concave polygon into triangles, parallelograms, or other forms whose areas are simple to find out.

Area of Concave Polygon = Area of the different shapes that are given

Some other formulas related to concave polygons are:

Formula	Description
A = 1/2 × n × s	Area of a regular concave polygon, where A is the area, n is the number of sides, and s is the length of a side.
A = 1/2 × (n − 2) × s × h	Area of a concave polygon using side length and height, where A is the area, n is the number of sides, s is the length of a side, and h is the height.
A = 1/2 × n × r2 × sin(360°/n​)	Area of a concave polygon using the radius of the circumscribed circle, where A is the area, n is the number of sides, r is the radius, and sin⁡sin is the sine function.

Difference between Concave and Convex Polygon

The key differences between concave and convex polygons are listed in the following table:

Feature	Concave Polygon	Convex Polygon
Definition	At least one interior angle is greater than 180 degrees.	All interior angles are less than 180°.
Shape	At least one part of the polygon “bulges” inward, creating a concavity.	No part of the polygon “bulges” inward.
Edges	Some edges point inward, toward the interior of the polygon.	All edges point outward, away from the interior of the polygon.
Convexity	Only portions of the polygon may be convex; overall, the polygon is concave.	Entire polygon is considered to be convex.
Properties	May have more complex geometrical properties due to concavities.	Typically simpler to analyze and work with.
Examples	Star polygons, irregular polygons with one or more indentations.	Regular polygons (e.g., equilateral triangle, square, pentagon).

Read More about Convex Polygon.

Summary: Concave Polygon

We can summarize the complete article into the following table:

Characteristic	Description
Definition	A polygon with at least one interior angle greater than 180°.
Diagonals	Can have at least one diagonal outside the polygon.
Example	A concave quadrilateral may have an interior angle greater than 180∘180∘, causing the shape to “cave in” towards its interior.
Sum of Interior Angles for Concave Polygon	(n-2) × 180°, where n is the side.
Sum of Exterior Angles for Concave Polygon	360°

Read More,

• Polygons
• Types of Polygons
• Area of Polygon
• Area of Regular Polygon

Examples of Concave Polygon

Example 1: Given sides of a concave polygon are 12 cm, 11 cm, 13 cm, 14cm. Find out its perimeter?

Solution:

All sides of Concave polygons are 12 cm, 11 cm, 13 cm, 14cm.

Perimeter of Concave Polygon = 12 + 11 + 13 + 14 = 50 cm

Therefore, The Perimeter of Concave polygon is 50 cm.

Example 2: Consider a concave polygon with 8 sides. Now calculate the sum of its interior angles.

Solution:

The formula for finding the sum of interior angles of a polygon:

Sum of Interior Angles = (n−2) × 180°

So Here, n = 8

Sum of Interior Angles = (8 – 2) × 180°

⇒ Sum of Interior Angles = 1080°

Example 3: Let find the angle x, in the figure given below, calculate the interior angle sum.

Solution:

Number of polygon sides = 5

Interior Angle Sum = (n-2)(180°)

⇒ Interior Angle Sum = (5 -2)(180°)

⇒ Interior Angle Sum = 3 (180°)

⇒ Interior Angle Sum = 540°

Therefore, the sum of given polygon interior angle sum is 540°.

Interior angle sum = 540°

x + x + 52° + 86° + 144° = 540°

⇒ 2x + 282° = 540°

After subtracting 282° from both sides

2x + 282° – 282° = 540° – 282°

⇒ 2x = 258°

Now, lets’ divide both sides by 2

2x/ 2 = 258°/2

⇒ x = 129°

Therefore, the value of x = 129°

FAQs on Concave Polygon

What is a concave polygon?

Concave polygons can also be known as non-convex polygons, with any one interior angle that measures greater than 180°.

How many sides does a concave polygon have?

Four sides are minimum for a concave polygon.

Is rhombus a convex polygon or a concave polygon?

A rhombus is a form of parallelogram in which is measured by the opposite angles and all of the sides are equal. Therefore, Rhombus is a convex polygon.

What is the Sum of Exterior Angles of a Concave Polygon?

A Sum of concave polygon’s exterior angles is add up to 360°. This means that all the concave polygon of exterior angles can be measured in 360°, like other polygons.

How to Find the Perimeter of a Concave Polygon?

The method used to find the perimeter of other concave polygons is calculated by adding all the sides of a polygon.

Is Square a Concave Polygon?

No, a square is a convex polygon rather than a concave polygon, as all of its diagonals fall inside of it and there isn’t a single interior angle that is greater than 180°.