<P> The radii of the incircle is related to the area of the triangle . The ratio of the area of the incircle to the area of the triangle is less than or equal to π 3 3 (\ displaystyle (\ frac (\ pi) (3 (\ sqrt (3))))), with equality holding only for equilateral triangles . </P> <P> Suppose △ A B C (\ displaystyle \ triangle ABC) has an incircle with radius r and center I. Let a be the length of BC, b the length of AC, and c the length of AB . Now, the incircle is tangent to AB at some point C ′, and so ∠ A T c I (\ displaystyle \ angle AT_ (c) I) is right . Thus the radius T I is an altitude of △ I A B (\ displaystyle \ triangle IAB). Therefore, △ I A B (\ displaystyle \ triangle IAB) has base length c and height r, and so has area 1 2 c r (\ displaystyle (\ tfrac (1) (2)) cr). Similarly, △ I A C (\ displaystyle \ triangle IAC) has area 1 2 b r (\ displaystyle (\ tfrac (1) (2)) br) and △ I B C (\ displaystyle \ triangle IBC) has area 1 2 a r (\ displaystyle (\ tfrac (1) (2)) ar). Since these three triangles decompose △ A B C (\ displaystyle \ triangle ABC), we see that the area Δ of △ A B C (\ displaystyle \ Delta (\ text (of)) \ triangle ABC) is: </P> <Dl> <Dd> Δ = 1 2 (a + b + c) r = s r, (\ displaystyle \ Delta = (\ frac (1) (2)) (a + b + c) r = sr,) and r = Δ s, (\ displaystyle r = (\ frac (\ Delta) (s)),) </Dd> </Dl> <Dd> Δ = 1 2 (a + b + c) r = s r, (\ displaystyle \ Delta = (\ frac (1) (2)) (a + b + c) r = sr,) and r = Δ s, (\ displaystyle r = (\ frac (\ Delta) (s)),) </Dd>

Largest circle that can fit in a triangle