A sphere having a radius of 6 cm rests on 3 horizontal wires forming a plane triangle whose sides are 5 cm, 12 cm and 13 cm. What is the height of the top of the sphere above the plane of the wires?
1. The tangent points of the sphere and the wires are placed on the incircle of the triangle. The incircle has the radius r. The sphere has the radius R.
2. The triangle is a right triangle because $\displaystyle 5^2+ 12^2 = 13^2$
3. Let A denote the area of the triangle. Then the radius of the incircle is calculated by:
$\displaystyle r = \dfrac {2A}{a+b+c} = \dfrac{ a \cdot b}{a+b+c}$
With your values you'll get $\displaystyle r = \dfrac{60}{30}=2$
4. The height of the sphere above the triangle is $\displaystyle h = R+h'$ with
$\displaystyle h'^2+r^2=R^2$
With your values you'll get: $\displaystyle h'=\sqrt{32} = 4\sqrt{2}$
and consequently $\displaystyle h = R+h'~\implies~h=6+4\sqrt{2}\approx 11.66$