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

Edited by Tayyaba Rehman — By Fiza Rafique — Published on November 22, 2023
Concave Polygons have at least one angle > 180°, while Convex Polygons have all angles < 180°.

## Key Differences

Concave Polygons present a unique geometric structure, offering an inward curve, or caving at least at one point on their perimeters. Convex Polygons, contrastingly, possess a structure where all its points on the perimeter point outward or bulge away. Both possess entirely different properties when it comes to angles, vertices, and real-world applications, emphasizing their uniqueness and importance in geometry.
In practical applications, Concave Polygons may not always fit well within certain mathematical models, such as computational geometry algorithms. Convex Polygons are often favored in these contexts due to their simpler, outward-bulging nature which ensures that any line segment drawn between any two points inside the polygon will always remain inside the polygon. This leads to the fact that mathematical and computer algorithms often find it easier to work with convex shapes.
When discussing interior angles, Concave Polygons display a variance as they possess at least one interior angle measuring more than 180 degrees. In contrast, Convex Polygons consistently showcase interior angles that are less than 180 degrees, establishing a straightforward, outward-projecting shape without any recesses, which can be easier to analyze in certain mathematical scenarios.
In computational applications, the distinction between Concave and Convex Polygons can be vital. Algorithms for processes like collision detection in video games often prefer Convex Polygons because of their straightforwardness in mathematical calculations. Whereas, dealing with Concave Polygons may require additional computational resources due to their inward-curving nature.
Navigating through real-world applications, Concave Polygons might be observed in designs or structures that require an indentation or an inward design aspect for aesthetic or functional purposes. Conversely, Convex Polygons might be observed in scenarios that necessitate a bulging outward feature, such as in certain architectural designs or objects, owing to their all-outward-pointing vertices.

## Comparison Chart

### Definition

Have at least one angle > 180°.
All angles are < 180°.

### Interior Angles

Contains at least one angle > 180°.
All angles are < 180°.

### Example Shape

A polygon that looks like it has a bite taken out of it.
A regular, outward-bulging polygon.

### Real-world Applications

Used where an inward curve is needed in design.
Preferable where robustness is desired.

### Computational Simplicity

Can be computationally complex to manage.
Typically simpler to calculate and manage.

## Compare with Definitions

#### Concave Polygons

Concave Polygons possess at least one diagonal that exists outside of the polygon.
In Concave Polygons, certain diagonals, when drawn, stretch outside the figure.

#### Convex Polygons

Convex Polygons have all interior angles less than 180 degrees.
Squares are Convex Polygons that are utilized in numerous grid designs.

#### Concave Polygons

Concave Polygons can have a portion of their perimeter curved inward.
Concave Polygons are evident in various artistic designs that require an inward curve.

#### Convex Polygons

Convex Polygons are computationally simpler to manage in many algorithms.
In collision detection, Convex Polygons are often preferred for simplicity.

#### Concave Polygons

Concave Polygons are not always the preferable choice in computational geometry.
For algorithmic simplicity, programmers sometimes avoid using Concave Polygons.

#### Convex Polygons

Convex Polygons ensure that any line segment between two points within them lies inside the polygon.
Convex Polygons are advantageous in 3D modeling due to their inherent simplicity.

#### Concave Polygons

Concave Polygons contain at least one vertex that is not part of the polygon’s hull.
The arrow shape is a classic example of Concave Polygons with an inward-pointing vertex.

#### Convex Polygons

Convex Polygons do not possess any vertex that’s pushed inward.
A regular hexagon, used often in board game designs, is a Convex Polygon.

#### Concave Polygons

Concave Polygons have at least one interior angle greater than 180 degrees.
The logo of a star often uses Concave Polygons in its design.

#### Convex Polygons

Convex Polygons ensure all diagonals are contained within the polygon.
Convex Polygons, like rectangles, ensure all diagonals remain inside.

## Common Curiosities

#### Are all angles in Convex Polygons less than 180 degrees?

Yes, all angles in Convex Polygons are less than 180 degrees.

#### Can Concave Polygons have all angles less than 180 degrees?

No, Concave Polygons must have at least one angle greater than 180 degrees.

#### Is it true that Convex Polygons have no vertices pushed inward?

Yes, Convex Polygons have all vertices bulging outward.

#### What is one primary visual difference between Concave and Convex Polygons?

Concave Polygons have at least one inward-pushing vertex, whereas Convex Polygons do not.

#### Are Convex Polygons computationally simpler to manage?

Yes, due to their straightforward geometric nature, Convex Polygons are often simpler computationally.

#### Why are Convex Polygons preferred in certain algorithms?

Convex Polygons are preferred for their geometric simplicity and all-internal diagonals.

#### What defines Concave Polygons?

Concave Polygons have at least one angle measuring more than 180 degrees.

#### Is it true that Concave Polygons might contain a diagonal outside the polygon?

Yes, Concave Polygons can have at least one diagonal lying outside the polygon.

#### Are star-shaped polygons considered Concave Polygons?

Yes, because star-shaped polygons have inward-pushing vertices, making them Concave Polygons.

#### Is it difficult to calculate the area of Concave Polygons?

It can be more complex due to the inward vertices which may require additional calculations.

#### Can a Concave Polygon be converted to a Convex Polygon?

Not without altering its vertices and hence changing its original shape.

#### Can Convex Polygons have an interior angle greater than 180 degrees?

No, all interior angles in Convex Polygons are less than 180 degrees.

#### Are squares considered Convex Polygons?

Yes, squares, having all angles less than 180 degrees, are Convex Polygons.