Sitting pretty

The cross-section of the sphere in the plane of the triangle will be the incircle of said triangle. In general, the inradius $r$ for a triangle of sides length $x,y,z$ is given by the formula

$r = \sqrt{\dfrac{(x + y - z)(x + z - y)(y + z - x)}{x + y + z}}.$

For the given triangle we find that $r = \dfrac{\sqrt{15}}{2}$ .

Now form a right triangle with vertices being the center of the incircle, the center of the sphere and any one of the points of tangency of the sphere with the metal triangle. Then the hypotenuse will be the radius of the sphere, one leg will be the inradius of the incircle and the other leg will have a length that represents the height of the center of the sphere above the plane of the triangle. By Pythagoras, the latter leg will have length

$\sqrt{4^{2} - (\dfrac{\sqrt{15}}{2})^{2}} = \dfrac{7}{2}$ .

To find $H$ we just need to add the radius of the sphere to this last value to find that

$H = 4 + \dfrac{7}{2} = \dfrac{15}{2}$ .

Thus $a + b = 15 + 2 = \boxed{17}$ .

Area of triangle=21√15\4 (by herons formula)

(6+7+8)r\2=area

r=√15\2

By Pythagoras theorem.

Distance of centre of sphere from plane of triangle=√(4^2-(√15\2)^2) =7\2

Height of top of sphere=4+7\2=15\2

15+2=17