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Incircle

Revision as of 14:14, 1 January 2014 by Claudeaops (talk | contribs)

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Triangle ABC with incenter I, with angle bisectors (red), incircle (blue), and inradii (green)

An incircle of a convex polygon is a circle which is inside the figure and tangent to each side. Every triangle and regular polygon has a unique incircle, but in general polygons with 4 or more sides (such as non- square rectangles) do not have an incircle. Quadrilaterals that do have an incircle are called tangential quadrilaterals.

Formulas

  • The radius of an incircle of a triangle (the inradius) with sides $a,b,c$ and area $K$ is $\frac{2K}{a+b+c}$
  • The radius of an incircle of a right triangle (the inradius) with legs $a,b$ and hypotenuse $c$ is $\frac{ab}{a+b+c}=\frac{a+b-c}{2}$.
  • For any polygon with an incircle, $K=sr$, where $K$ is the area, $s$ is the semiperimeter, and $r$ is the inradius.
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