# Concave Polygons vs. Convex Polygons: What's the Difference?

Edited by Harlon Moss || By Janet White || Published on January 27, 2024
Concave polygons have at least one interior angle greater than 180° and inward indentations; convex polygons have all interior angles less than 180°, forming a bulged shape.

## Key Differences

Concave polygons exhibit at least one interior angle exceeding 180 degrees, resulting in an inward curve. In contrast, convex polygons feature interior angles that are all less than 180 degrees, creating a consistently outward-facing edge without any indentations.
Concave polygons can appear to have a 'caved-in' section due to their inward-facing vertices. Convex polygons, on the other hand, bulge outward, forming a shape where any line segment between two points on the polygon lies entirely inside it.
In concave polygons, one or more interior angles are greater than 180 degrees, and they can have vertices pointing inward. Conversely, convex polygons have all interior angles less than 180 degrees and lack inward-pointing vertices.
Concave polygons are often found in irregular shapes like star polygons, while convex polygons are more common in regular shapes like squares and triangles. This distinction is important in fields like architecture and computer graphics.
The mathematical characterization of concave polygons involves the presence of non-adjacent vertices, from which a line can be drawn that passes outside the polygon. For convex polygons, any such line will always remain inside the figure.

## Comparison Chart

### Interior Angles

At least one angle > 180°
All angles < 180°

### Edge Direction

Can have inward indentations
Edges bulge outward

### Line Segment Rule

A line between points may lie outside the shape
All lines between points stay inside

### Vertex Orientation

Can have vertices pointing inward
No vertices pointing inward

### Practical Examples

Star-shaped polygons
Regular shapes like squares and triangles

## Concave Polygons and Convex Polygons Definitions

#### Concave Polygons

A polygon not containing the line segment between any two points within it.
A polygon where some internal lines intersect its edges is concave.

#### Convex Polygons

A polygon where the line segment between any two points does not extend outside the shape.
A triangle, with no line extending outside, exemplifies a convex polygon.

#### Concave Polygons

A polygon where a line segment between two points on the polygon can pass outside it.
In a concave polygon, drawing a line between certain points may cross outside the shape.

#### Convex Polygons

A polygon where any line segment between two points lies entirely inside it.
In a convex polygon like a rectangle, all internal lines remain inside.

#### Concave Polygons

A polygon that appears to have a 'caved-in' section.
A polygon resembling a crescent moon is concave.

#### Convex Polygons

A polygon with edges that bulge outward.
A circle approximated by a polygon is convex.

#### Concave Polygons

A polygon with one or more inward-pointing vertices.
A polygon with an inward dent forms a concave shape.

#### Convex Polygons

A polygon with no vertices pointing inward.
A square, with all outward-facing corners, is a convex polygon.

#### Concave Polygons

A polygon with at least one interior angle greater than 180°.
A star-shaped polygon is a typical concave polygon.

#### Convex Polygons

A polygon with all interior angles less than 180°.
A regular hexagon is a perfect example of a convex polygon.

## FAQs

#### Are the edges of convex polygons always outward-facing?

Yes, convex polygons have outward-facing edges without any indentations.

#### Can a concave polygon have a 'caved-in' appearance?

Yes, concave polygons often have a 'caved-in' appearance due to inward-pointing vertices.

#### What defines a concave polygon?

A concave polygon has at least one interior angle greater than 180° and inward indentations.

#### What happens to a line drawn between two points in a concave polygon?

In a concave polygon, such a line can sometimes pass outside the polygon.

#### Can a line segment inside a concave polygon lie outside the polygon?

Yes, in concave polygons, some line segments can extend outside the polygon.

#### What is a convex polygon?

A convex polygon has all interior angles less than 180°, with no indentations.

#### Is a square a concave or convex polygon?

A square is a convex polygon as all its interior angles are less than 180°.

#### Can concave polygons form regular shapes?

Generally, concave polygons do not form regular shapes due to their indentations.

#### What is an example of a concave polygon?

A star-shaped polygon is an example of a concave polygon.

#### Can convex polygons have any inward-pointing vertices?

No, convex polygons do not have any inward-pointing vertices.

#### Are triangles considered convex polygons?

Yes, triangles are convex polygons as all their interior angles are less than 180°.

#### Is it possible for a convex polygon to have an inward dent?

No, convex polygons cannot have inward dents.

#### Is a rectangle an example of a convex polygon?

Yes, a rectangle is a convex polygon as it adheres to the angle criteria.

#### Do concave polygons have more complex mathematical properties than convex ones?

Yes, concave polygons often have more complex mathematical properties due to their inward indentations.

#### Are all regular polygons convex?

Yes, all regular polygons, like squares and equilateral triangles, are convex.

#### Do convex polygons have any interior angles greater than 180°?

No, convex polygons have all interior angles less than 180°.

#### Are concave polygons common in architecture?

Concave polygons are less common in architecture due to their complex shapes.

#### What is a distinguishing feature of concave polygons?

A distinguishing feature of concave polygons is their inward-pointing vertices or edges.

#### Can convex polygons be irregular in shape?

Convex polygons can be irregular, but all their angles must still be less than 180°.

#### Are star-shaped polygons typically concave or convex?

Star-shaped polygons are typically concave due to their inward angles.