»  Solutions to puzzles in my National Review Online Diary

  January 2006


In my January diary I posed the following brain-teaser.

There is a ball 12 feet in diameter on top of a pole 60 feet high. On the ball stands a man whose eye is 6 ft above the ball. How much ground beneath the ball is invisible to him?



Too easy. The man's eye is at the apex of a triangle 78 feet high (60 + 12 + 6). A tangent to the sphere from this point makes an angle of 30 degrees with the vertical. (Easy, by construction: the lesser triangle made by the eye, the sphere's center, and the tangent point, is right-angled, with hypoteneuse 12 ft and short side 6 ft.) That makes the main triangle, height 78 ft, equilateral — all angles 60 degrees, height to base in the ratio √3 to 2. Half the triangle's base is therefore 78 / √3, which is equal to 26√3. The ground invisible to the man is a circle with this radius, area 2028π.