For 1D, 1×1 means the place of 1/8 C, for 2D, it's the place of 1/4 A.

So the better equation is:

A = 0.7071*0.7071 + 0.2929*1 or 0.7071*1 + 0.2929*0.2929.

By this 2D method, you can see two methods with the identical result, i.e. the whole circumference area equals all 1×1 space minus a 'yard' as broad as 0.7071*0.2929. (=0.2071). So A = 1-0.2701=0.7929.

This area if changed to be a perfect round area will have the same size: 0.7929. (r=0.5).

This is the area for special angle: 90°. While other angles have complicated and or non linear calculations. The comprehensive equation has not been found yet.