Definition: A circle inside a triangle or regular polygon
that touches every side of it at one point.

In the case of a triangle, there is always an incircle possible, no matter what shape the triangle is. In the figure above, the red circle is the incircle of the triangle.

- Each side of the triangle is a tangent to the incircle.
- The center of the incircle is called the incenter. See Incenter of a Triangle

It is possible to construct the incircle with a compass and straightedge. See Constructing the Incircle and Incenter of a Triangle

Regular polygons, (polygons that have all sides the same length and all interior angles congruent) can have incircles. As with the triangles case, each side of the polygon is a tangent to the incircle.

The center of the incircle, the incenter, is also considered to be the center of the polygon itself, since it is equidistant from each vertex.

For more on this see Incircle of a Regular Polygon and Regular Polygon definition.

Irregular polygons are not thought of as having an incircle or even a center. If you were to draw a polygon at random, it is unlikely that there is a circle that has every side as a tangent. An exception is the 3-sided polygon (triangle). All triangles always have an incircle (see above).

- Circle definition
- Radius of a circle
- Diameter of a circle
- Circumference of a circle
- Parts of a circle (diagram)
- Semicircle definition
- Tangent
- Secant
- Chord
- Intersecting chords theorem
- Intersecting secant lengths theorem
- Intersecting secant angles theorem
- Area of a circle
- Concentric circles
- Annulus
- Area of an annulus
- Sector of a circle
- Area of a circle sector
- Segment of a circle
- Area of a circle segment (given central angle)
- Area of a circle segment (given segment height)

- Basic Equation of a Circle (Center at origin)
- General Equation of a Circle (Center anywhere)
- Parametric Equation of a Circle

- Arc
- Arc length
- Arc angle measure
- Adjacent arcs
- Major/minor arcs
- Intercepted Arc
- Sector of a circle
- Radius of an arc or segment, given height/width
- Sagitta - height of an arc or segment

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