If the lengths of the sides of a triangle are $3, 4$ and $5$ units,then $R$ (the circumradius) is ............ $unit$.

If $R$ is the radius of the circumcircle of the $\Delta ABC$ and $\Delta$ is its area,then

In a $\triangle ABC$,$2A + C = 300^{\circ}$. If the circumradius $R$ of the $\triangle ABC$ is eight times its inradius $r$,then $\sin \frac{C}{2} = $

Let $ABC$ be a scalene triangle with incentre $I$ and circumcentre $O$. Suppose $B, C, I, O$ are concyclic points. Then $\angle B + \angle C$ is (in $^{\circ}$)

If the roots of the equation $x^3-11x^2+36x-36=0$ are the ex-radii of a triangle $ABC$, then the perimeter of the triangle $ABC$ is

Let $ABC$ be an equilateral triangle with side length $a$. Let $R$ and $r$ denote the radii of the circumcircle and the incircle of triangle $ABC$ respectively. Then,as a function of $a$,the ratio $\frac{R}{r}$