Inradius
The inradius of a polygon is the radius of its incircle (assuming an incircle exists). It is commonly denoted 
.
![[asy] pathpen = linewidth(0.7); pair A=(0,0),B=(4,0),C=(1.5,2),I=incenter(A,B,C),F=foot(I,A,B); D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle); D(CR(D(MP("I",I,SW)),inradius(A,B,C))); D(F--I--foot(I,B,C)--I--foot(I,C,A)); D(rightanglemark(I,F,B,5)); MP("r",(F+I)/2,E); [/asy]](http://latex.artofproblemsolving.com/3/a/d/3ad1b30f1350fc22f90ce7a79f2567025d4fe4c0.png)
Properties
- If
has inradius 
and semi-perimeter 
, then the area of is 
. This formula holds true for other polygons if the incircle exists. - The in radius satisfies the inequality
, where 
is the circumradius (see below). - If has inradius and circumradius , then
.
has inradius 
, then the 
. This formula holds true for other polygons if the incircle exists.
, where 
is the 
.Problems
- Verify the inequality
. - Verify the identity (see Carnot's Theorem).
- 2007 AIME II Problems/Problem 15
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