SOLUTION: A rectangle ABCD has dimensions AB = a and BC = b. Four semi-circles, facing outward, are drawn using each edge of the rectangle as diameter. A full circle is drawn using the diag

Algebra ->  Surface-area -> SOLUTION: A rectangle ABCD has dimensions AB = a and BC = b. Four semi-circles, facing outward, are drawn using each edge of the rectangle as diameter. A full circle is drawn using the diag      Log On


   

Question 1071414: A rectangle ABCD has dimensions AB = a and BC = b. Four semi-circles, facing outward, are drawn using each edge of the rectangle as diameter. A full circle is drawn using the diagonal of the rectangle as diameter. Find the total area of the “4 lunes”, the shaded area that is inside one of the first
4 semicircles circles but outside the full circle.
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Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.
2%2A%281%2F2%29%2A%28%28pi%2Aa%5E2%29%2F4%29 + 2%2A%281%2F2%29%2A%28%28pi%2Ab%5E2%29%2F4%29 - pi%2A%28a%5E2%2Bb%5E2%29%2F4 = 0.

In the left side the first two addends are the areas of the four semi-circles built on the sides of the rectangle.

The subtrahend is the area if the big circle.


Similar to that as 
     "the area of the square built on the hypotenuse of the right-angled triangle is equal to the sum of areas 
     of two squares built on its legs"

(Pythagoras), 

     "the area of the semi-circle built on the hypotenuse of the right-angled triangle is equal to the sum of areas 
      of two semi-circles built on its legs"

Answer. 0. Zero. ZERO.