Suppose you have a circular pool of lava (the reason for the contents will be clear in a moment) and in the center of the pool is a circular lawn. With a single straight-line measurement, determine the area of the
The measuring instrument you have is a laser transit, with which you can look through a telescope, point it to a pole your assistant is holding, and read off the distance from the transit to the pole. Because you and your assistant can't stand in the lava, you can't use the obvious strategy of measuring the diameter of the lawn, but you can find the distance between any two points outside of the lava. How can you find the area of the
An apology is in order here. I was apparently passing a brain stone when I posted the original question. I should have asked what the area of the lava was. The comments have already answered the question.
My real purpose was to observe that measuring the chord will give you the area of the lava, regardless of what the diameter of the pool and the lava were, so you could shrink the lawn diameter to zero and the chord would allow you to compute the area of the lava. In this way, the problem is like the question spatial geometry hole in sphere.
My eventual question was to be, does anyone know of similar problems, where the answer doesn't require one of parameters of the problem and so can be solved by setting that parameter to zero?