Random point on a snowman below 39 inches

A snowman is a stack of three spheres touching end to end, with radii $14$ (bottom), $11$ (middle) and $8$ (top) inches, the bottom one resting on the ground. Treat the whole surface of all three spheres as exposed, and choose a point uniformly from that total surface. What is the probability the point is no higher than $39$ inches above the ground?

$171/254$. By Archimedes' hat-box theorem, a horizontal slice of a sphere has area proportional to its height. The bottom sphere (height $0$–$28$) counts fully, and $39$ is exactly the middle sphere's equator, so half of it counts: $(4\cdot196+2\cdot121)\pi=1026\pi$ out of $4(64+121+196)\pi=1524\pi$, giving $\tfrac{171}{254}$.