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From “God Plays Dice,” an interesting geometry problem

A clever geometry problem:

Here's a cute problem (from Robert M. Young, Excursions in Calculus, p. 244): "What is the average straight line distance between two points on a sphere of radius 1?"

(Answer to follow.)

Note the number of comments with different answers.

"Excursions in Calculus," from what I could see at Google Books, looks like it has many interesting problems.

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  1. square root of 2?

  2. Integral of 2cosxsin2x x varying from 0 to 90 degrees
    = -4t^2dt t varying from 1 to 0
    = 4/3

  3. I have games on my blog much simpler if you want to solve some, appear.
    Greetings pekota.

  4. The answer is 2/3, being the average chord length between any 2 points on a circle of radius 1.

  5. This is a quite fun problem indeed.

    1/4π ∫[0->π] (2π sinθ)(2sinθ/2) dθ = 4/3

    Here’s my blog post explaining why this is so:

    Have you guys thought what is the average straight line distance between two points on an unit n-sphere?

  6. I found this enlightening:


  7. The answer is the volume divided by the area of the sphere:

    (4/3 π r³) / (2 π r²) = 2 / 3 r

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