by Guest » Sun Feb 02, 2014 4:19 pm
Just a simple solution....
You need to observe that there is a lot of symmetry in this construction.
If we join the curve intersection points we will form chords and segments of
the arcs and these will also form a square with straight sides. The area we
want to find will be the area of this square plus the area of the 4 segments.
The angle subtended by this chord to the corner of the large 4 unit square
will be 30 degrees.
The length of the chords will be = 2 x 4 x Sin 15 = 2.071 units
Area of the square part will be 2.071 x 2.071 = 4.287 sq units
Area of the segments = ((Pi x 4 x 4) / (12) = 4/3 x Pi) - Area of triangle
Area of triangle = 0.5 x 4 x 4 x Sin 15 = 4 sq units
So area of segment = ((4/3 x Pi) - 4) = 0.189
total for 4 of them = 0.757 sq units
So Area of curved Square 4.287 + 0.757 = 5.044 Sq Units.
.................Simple.............