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Concave Polygon: Definition & Examples

Instructor: Joseph Vigil
In this lesson, review what polygons are. Then, learn about the angle and resulting qualities that make concave polygons unique. Afterward, you can test your knowledge with a brief quiz.

Polygons

A polygon is an enclosed two-dimensional shape with straight sides. So many of the shapes we're familiar with, such as triangles, squares, and rectangles are polygons because they're enclosed shapes and have straight sides.

Straight sides make a polygon.
Straight sides make a polygon.

Circles and ovals, however, aren't polygons because their exteriors consist of one large curve. Although they're enclosed shapes, they don't have the straight sides required of a polygon.

No straight sides, no polygon
No straight sides, no polygon

Concave Polygons

A concave polygon is a polygon that has at least one angle greater than 180 degrees. In other words, it has at least one angle that extends beyond a straight line. Because of this angle, at least one of the vertices, or corners, will appear pushed in.

Consider this concave polygon, for example:

triangle with right side pushed in

See how the right side of this would-be triangle appears pushed in? That's because of the angle at that vertex measuring more than 180 degrees.

concave polygon with angle indicated

Another way to think of concave polygons is that you can create a 'cave' by enclosing them with an exterior line. In our previous example, we could enclose the polygon at the 'pushed-in' side:

enclosed concave polygon

The red line encloses the polygon, creating a 'cave'.

This polygon is also concave:

concave polygon

Here's its angle larger than 180 degrees:

concave polygon with largest angle indicated

Because it's concave, we could enclose this polygon with an exterior line:

enclosed concave polygon

Here's one more example of a concave polygon:

concave polygon

Can you spot the angle that makes this polygon concave? It's there at the bottom because that angle extends past a straight line:

concave polygon with the largest angle indicated

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