A surprise in area and arc-length

 

Consider the first quadrant of a unit circle. Let s be any arc. Then the area between s and the x-axis plus the area between s and y-axis is a constant!

 

If we forget about the unit, then           A + B = s

    Note that the regions A and B overlap and that portion of area is counted twice. (in brown)

 

 

 

 


Let P(x1, y1), Q(x2, y2)

The circle is         x2 + y2 = 1

      

 

      

\ s = A + B

 

 

Sector Fun

Let ÐPOR = a,  ÐQOR = b in radians.

 

Then  s = b - a.

 

OP = OQ = 1

A = sector POQ + DPOR - DQOS

                                               (equation 1)

 

B = sector POQ + DQOT - DPOU

          (equation 2)

Adding equations 1 and 2, we get:

               A + B = a + b = s

 

 

Final Note :

 

       Since the radius is 1, the two methods above are equivalent because: