# Incenter of a regular polygon

The point where the interior angle bisectors intersect.
Try this Drag any orange dot and change the number of sides. Note where the bisectors of the interior angles meet.

If you bisect the interior angles of a regular polygon, the bisectors will always converge at the same point - called the incenter of the polygon. The incenter is also the center of:

1. The incircle - the largest circle that will fit inside the polygon
2. The circumcircle - the circle that passes through every vertex.

For irregular polygons - where the side lengths and interior angles are all different - the angle bisectors do not meet at a single point, so irregular polygons have no incenter.

### Except triangles!

All triangles have an incenter, regardless of shape and proportions. See Incenter of a Triangle.

## Other centers

Regular polygons (and all triangles) have other centers too:

• Circumcenter - the point where the three perpendicular bisectors of the sides meet.
• Centroid - the point where the altitudes meet.
Note however that in the case of regular polygons these centers are all in the same place. In the case of triangles they are usually in different places. (In equilateral triangles, which are essentially three-sided regular polygons, they are in the same place).

While you are here..

... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone. However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site? When we reach the goal I will remove all advertising from the site.

It only takes a minute and any amount would be greatly appreciated. Thank you for considering it!   – John Page

Become a patron of the site at   patreon.com/mathopenref