In a triangle whose sides are $18$, $24$ and $30 \; cm$ respectively, find the circumradius, the inradius and the radii of the three escribed circles.
Let $ABC$ be the given triangle with sides $\;$ $a = 18 \; cm$, $\;$ $b = 24 \; cm$, $\;$ $c = 30 \; cm$
Semi-perimeter of $\triangle ABC$ $= s = \dfrac{a + b + c}{2} = \dfrac{18 + 24 + 30}{2} = 36 \; cm$
Area of $\triangle ABC$ $= \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$
i.e. $\;$ $\Delta = \sqrt{36 \left(36 - 18\right) \left(36 - 24\right) \left(36 - 30\right)}$
i.e. $\;$ $\Delta = \sqrt{36 \times 18 \times 12 \times 6} = 216 \; cm^2$
Circumradius $R = \dfrac{abc}{4 \Delta} = \dfrac{18 \times 24 \times 30}{4 \times 216} = 15 \; cm$
Inradius $r = \dfrac{\Delta}{s} = \dfrac{216}{36} = 6 \; cm$
Escribed radius $r_1 = \dfrac{\Delta}{s - a} = \dfrac{216}{36 - 18} = 12 \; cm$
Escribed radius $r_2 = \dfrac{\Delta}{s - b} = \dfrac{216}{36 - 24} = 18 \; cm$
Escribed radius $r_3 = \dfrac{\Delta}{s - c} = \dfrac{216}{36 - 30} = 36 \; cm$