In an equilateral triangle,the inradius $(r)$ and the circumradius $(R)$ are connected by which of the following relations?

If the lengths of the sides of a triangle are $15, 20, 25$ units,find the circumradius of the triangle. (in $units$)

In a triangle $ABC,$ $a:b:c = 4:5:6$. The ratio of the radius of the circumcircle to that of the incircle is

If $\triangle ABC$ is right-angled at $A$, then $r_2+r_3$ is equal to

In $\triangle ABC$,suppose the radius of the excircle opposite to angle $A$ is denoted by $r_1$,similarly $r_2$ for angle $B$,and $r_3$ for angle $C$. If $r$ is the radius of the inscribed circle,then what is the value of $\frac{ab - r_1 r_2}{r_3}$?

In $\Delta ABC$,$AB = AC$. Let $P_1$ denote the incircle of $\Delta ABC$. Circle $P_2$ is tangent to sides $AB$,$AC$ and to circle $P_1$. If the radii of circles $P_1$ and $P_2$ are $2$ and $1$ respectively,then the area of $\Delta ABC$ is: