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Concave polygon

 

A concave polygon is a polygon that has at least one angle greater than 180 degrees. This means if one angle inside a polygon is between 180 degrees and 360 degrees, then the polygon is concave. In addition, it must have a minimum of four sides and must be an enclosed figure. If the polygon has an angle greater than 180 degrees, but it is not an enclosed figure, then that polygon is not termed as a concave polygon.

A non-concave polygon is known as a convex polygon. The shape of a concave polygon is irregular. The sum of angles inside a polygon can be anything. It can be greater than 360 degrees as well. Therefore, there must be a reflex angle inside a concave polygon. The polygon can be constructed both inward and outward.

A three-sided polygon will not result in any concave polygon as the sum of angles of a three-sided polygon, i.e., a triangle is already equal to 180 degrees. And for a concave polygon, we need to have one angle greater than 180 degrees. Therefore, a triangle cannot be a concave polygon.

Regular and irregular concave polygon

•  Regular concave polygon: A regular polygon has all sides, and all interior angles are equal.

From the definition of a concave polygon, we know, one of its angles must be greater than or equal to 180 degrees.

And also, the sum of angles inside a polygon is given by (n – 2) x 180 degrees, where n = the number of sides of a polygon.

We can conclude from here that this condition cannot be satisfied at any cost. It is impossible to have a concave polygon with all sides equal and all angles equal. Therefore, a regular concave polygon does not exist.

•  Irregular concave polygon: We can easily find irregular polygons. In irregular polygons, the sides and angles can be of varying numbers. Just the condition of concavity must be there, i.e., one angle should be reflex. The sum of interior angles is of varying units in an irregular concave polygon. In conclusion, all concave polygons are irregular polygons.

Area and perimeter of concave polygon

We can find the area and perimeter by splitting the polygon into two or more shapes.
For example, consider the shape given in the figure.

We need to find the area and perimeter of the given concave polygon. Let us split this figure into different shapes. We will have one square and one rectangle after the split, as shown in the figure.

The area of first shape, i.e. square = 8 x 8 = 64 square units.
The area of second shape, i.e. rectangle = 10 x 24 = 240 square units.
Therefore, the total area of the polygon = 304 square units.

Now let us find the perimeter of the concave polygon. Perimeter = the sum of all sides of the polygon. Therefore,

The sum of all sides of a polygon = 8 + 8 + 16 + 10 + 24 + 10 + 8 = 84 units.

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