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Parcelation (Posted on 2018-10-18) Difficulty: 3 of 5
The square ABCD has side length 2*sqrt2.
A circle with centre A and radius 1 is drawn. A second circle with centre C is drawn so that it just touches the 1st circle at point P on AC .

Determine the total area of the regions inside the square but outside the two circles.

Source: one of Mayhem problems

No Solution Yet Submitted by Ady TZIDON    
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soln | Comment 1 of 2
The large circle around C has radius CP = CA-1 = sqrt 2 (2 sqrt 2) -1 =3  
and, say, intersects AB at x and AD at y.
cos(<xCB) = (2 sqrt 2)/3, and thus
<xCB = 19.471 deg = <yCD  (congruent triangles)
<xCy = 90 deg - ( <xCB + <yCD ) = 51.058 deg
Area of sector xCy = (<xCy/360) (Area of large circle)
Area of sector xCy = (51.058/360) (pi 3^2) = 4.01005 sq units

Note: xB = yD = sqrt [3^2 - (2 sqrt 2)^2] = 1, so the sum of area of the two triangles xCB and xCD 
= 2 (1/2 base x height) 
= 1 x 2 sqrt 2 
= 2.828 sq units    


Area inside Square but outside sectors and two triangles
= Area of Square - (1/4 small circle area + sector xCy area + area of the two triangles xCB and yCD) 
(2 sqrt 2)^2 - (pi/4 +4.01005+ 2.828) = 8 - (0.785 + 4.01005 + 2.828)
= 0.3761 sq units

Edited on October 19, 2018, 3:04 am
  Posted by Steven Lord on 2018-10-18 10:22:24

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