Q.Find the inradius $r$ of the triangle whose sides are $a=13$, $b=14$, $c=15$.

Incircle and Inradius

Every triangle has exactly one circle that sits inside it and touches all three sides — the incircle, centred at the incentre $I$ (the meeting point of the three internal angle bisectors), with radius $r$, the inradius. Three theorems connect $r$ to the triangle's other measurements:

Why three different formulas for the same $r$?

Each connects $r$ to a different set of "known" quantities: $r=\Delta/s$ is the definitional one (area divided by semi-perimeter — think of the incircle as the largest circle that "fits" given the perimeter budget), $r=(s-a)\tan\frac A2$ ties $r$ to one vertex's angle and the tangent length from that vertex, and $r=4R\sin\frac A2\sin\frac B2\sin\frac C2$ expresses $r$ purely through the circumradius and the three angles — useful when no side lengths are given at all.

The key geometric fact behind $r=(s-a)\tan\frac A2$

The incircle touches side $AB$ at a point $P$ and side $AC$ at a point $Q$, and it is a standard (separately provable) fact that $AP=AQ=s-a$ — i.e. the tangent length from any vertex depends only on the semi-perimeter and the opposite side. Since $AI$ bisects angle $A$, the right triangle $AIP$ has angle $\frac A2$ at $A$, adjacent side $s-a$, and opposite side $r$ — so $\tan\frac A2=r/(s-a)$.

Why it matters

Beyond "find the inradius," this circle is the natural home for several classic identity-proof questions relating $r$ to the exradii $r_1,r_2,r_3$ — e.g. $\frac1{r_1}+\frac1{r_2}+\frac1{r_3}=\frac1r$ and $rr_1r_2r_3=\Delta^2$, both provable in a couple of lines once $r=\Delta/s$ and $r_i=\Delta/(s-\text{side}_i)$ are known.

A common trap

Students sometimes use $s$ (the semi-perimeter) where the formula needs $s-a$, or vice versa. Since both appear throughout this chapter, it is worth a mental check: $r=\Delta/s$ has $s$ alone in the denominator, while every exradius formula has $s-(\text{that vertex's opposite side})$ instead.

Worked micro-example

For $a=13,b=14,c=15$: $s=21,\Delta=84$, so $r=84/21=4$.

Questions on "incircle and inradius formula class 11 maths" and "inradius of a triangle important questions" are staple entries in CBSE board exam papers and JEE Main practice sets, since the Properties of Triangles chapter of the NCERT-aligned Class 11 Mathematics curriculum tests exactly this cluster of $r=\Delta/s$-style identities. Mastering all three inradius formulas together is what typically separates a quick full-marks answer from a stuck one under exam conditions.