find the probability that a randomly chosen point in a circle with radius r, lies in the inscribed regular octagon.

Question

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Answer 1

The area of a circle:

     A_c = πr²

 

Area of inscribed octagon:

     A_O = 2r²√2

 

Probability:  Area of octagon / area of circle

     P =  (2√2)r² / (πr²)

     P =  2√2 / π

     P = 0.900 (rounded)

Note that the probability (that is, ratio of the areas) is not dependent on the radius r.

 

Note:  Area of "kite" (one-fourth of octagon) is dw/2 = (r)(r√2)/2.

    There are four "kites" in an octagon, so A_O = 4*(r)(r√2)/2 = 2r²√2

 

 

Answer 2

Probability = (Area of Octagon)/(Area of Circle)

• Area of Circle = pi*r²
• Area of Octagon = 2a²( 1 + √2)  where a = the side length of the octagon.  If you know r but not a, then a = 2r·cos(67.5°) ≈ 0.765·r