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A unit square is rotated 45 degrees counterclockwise about one of its vertices, A.  What is the green area? May 14, 2020

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May 14, 2020
edited by hugomimihu  May 14, 2020
#2
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Connect A  to to top left  vertex  of the rotated square....call this point B

And call the top right  vertex of the right triangle formed = C

Then the area of this right triangle ABC = 1/2  area of the square  = 1/2

The other small green area will be a 45 - 45 - 90  right triangle  with  legs  of diagonal lengthof the square - the side length of the square =   (√2 - 1)

So  the  area of this triangle  = (1/2)(√2 - 1)^2  = (1/2) ( 2 + 1 - 2√2)  =  (1/2) (3 - 2√2)

So...the green area =   1/2  + (1/2) ( 3 - 2√2)  = (1/2) ( 1 + 3 - 2√2 ) = (1/2) (4 - 2√2)  =

(2 - √2)  units^2   May 14, 2020