In this chapter we will learn about the concept of concave polygons with properties and examples.

Let us first understand the concave shape.

## What is Concave shape?

**A shape that points inwards** is known as **Concave shape**.

When the two sides are pushed inwards you will get a concave shape.

Observe the above image.

Note the sides EF and ED and check how the sides are pointing inwards.

Hence, the hexagon is a concave polygon.

## What is Concave Polygon ?

The polygon in which **one of the internal angle measures greater than 180 degree** is called concave polygon.

The concave polygon shows** following features**;

⟹ one of the internal angle is greater than 180 degree.

⟹ one set of sides point inwards

⟹ the diagonals may extend outside the polygon shape.

⟹ the line joining any two point inside the polygon extend outside the given shape.

I hope you understood the basic concepts. Let us now look at examples of important concave polygons.

## Examples of Concave Polygon

Concave Quadrilateral

The quadrilateral in which one of the side measures greater than 180 degree angle is a concave quadrilateral.

Observe the below image.

Note that internal angle C = 229 degree. Hence, quadrilateral ABCD is concave polygons.

**Diagonal Test**

In the given quadrilateral, the diagonal DB lie outside the polygon shape.

Hence, ABCD is a concave quadrilateral.

**Two Point Test**

In the given quadrilateral, the line joining points F & G extend outside the polygon shape. Hence, the given polynomial is concave quadrilateral.

### Concave Pentagon

The pentagon in which one of the interior angle measures greater than 180 degree is a concave Pentagon.

In the above pentagon ABCDE, the internal angle E measures 228 degree. Hence the given polygon is a Concave Pentagon.

**Side Test**

In the given Pentagon, the sides EA and ED points inwards, hence it is a concave polygon.

**Diagonal Test**

In the given pentagon ABCDE, the diagonals extend outside the polygon’s body. Hence, the given polygon is concave in shape.

**Two point test**

The line joining two points inside the pentagon extend outside the polygon’s shape. Hence, the given shape is concave pentagon.

### Concave Hexagon

Hexagon is a polygon with 6 sides.

If any of the internal angle measure greater than 180 degree, then the hexagon will be of concave shape.

In the given hexagon FGHIJK, the internal angle measures 234 degree. Since the internal angle is greater than 180 degree, the polygon is concave hexagon.

**Side Test**

The side KJ and JI point inwards which is a sign of concave shape.

**Diagonal Test**

In the given hexagon, the diagonal KI extends outside the polygon’s shape. Hence the given image is concave hexagon.

**Two point test**

In the given hexagon, the line joining two points extend beyond the boundary of given shape. Hence, it is a concave hexagon.

### Concave Heptagon

Heptagon is a polygon with 7 sides.

In the given heptagon ABCDEFG;

∠E = 281 degree

∠G = 209 degree

Since both the angles are greater than 180 degree, the given shape is a concave heptagon.

**Side test**

In the given heptagon two set of sides AG-GF and FE-ED point inwards which is a sign of concave polygon.

**Diagonal Test**

Diagonal AF extend outside the polygon body. Hence the given shape is a concave heptagon.

**Two point test**

In the given heptagon, the line joining two point extend outside the heptagon body. Hence the given shape is concave heptagon.

### Concave Octagon

Octagon is a polygon with 8 sides.

Given below is the octagon ABCDEFG with ∠F = 238 degree.

Since one of the internal angle is greater than 180 degree, the given shape is a concave octagon.

**Side Test**

Here the sides GF and FE point inwards. This tells us that the shape is concave in nature.

**Diagonal Test**

Here the diagonal GE lies outside the given shape. Hence the given octagon is concave in nature.

**Two point test**

The line joining the two inside point is extended outside the given shape. Hence the polygon is concave octagon.