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VOOZH | about |
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VOOZH | about |
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.
Table of Content
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.
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°).
A concave polygon can be distinguished from other polygons by a few unique characteristics or properties;
Concave polygons can be categorised as:
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.
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.
Like any 2d geometric object, polygons can have interior as well as exterior angles.
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°.
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°
For any polygon, we can calculate
Let's discuss the formula for perimeter and area od concave polygons.
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
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 sinsin is the sine function. |
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.
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,
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.
👁 Solved-Examples-on-Angles-in-Concave-Polygons
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°
