Average area of inscribed triangle
Find the average area of an inscribed triangle in the unit circle.Assume that each vertex of the triangle is equally likely to be at any point of the unit circle and that the location of one vertex does not affect the likelyhood the location of another in any way.
(note that the maximum area is achieved by the equilateral triangle, which has side length sqrt(3) and area 3sqrt(3)/4. How does the maximum compare to the average?)
(there is a Hint: In order to reduce the problem to the calculation of a double integral, place one of the vertices of the triangle at (1,0), and use the polar angles teta1 and teta2 of the two other vertices as variables.What is the region of integration?)
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Solution Summary
This shows how to find the average area of an inscribed triangle in the unit circle.