After some searching, I wasn’t able to find a general equation for this, so I’ll have to fall back on calculus.

Ok, center the 5.25" circle at the origin, and the 5" circle with its center shifted to the right at (2.5 , 0 ). This makes the graphs of the two upper semicircles:

- y = sqrt( 5.25[sup]2[/sup] - x[sup]2[/sup] )
- y = sqrt( 5[sup]2[/sup] - ( x - 2.5 )[sup]2[/sup] )

Set them equal to each other to find the intersection:

27.5625 - x[sup]2[/sup] = 25 - x[sup]2[/sup] + 5x - 6.25

5x = 27.5625 - 25 + 6.25 = 8.8125

x = 1.7625

Now find the area under the larger semicircle by integrating

sqrt( 5.25[sup]2[/sup] - x[sup]2[/sup] )

for x in [-5.25, -2.5] (range not covered by the small circle).

Then, for x in [-2.5, 1.7625], find the area under the larger circle and above the smaller:

sqrt( 5.25[sup]2[/sup] - x [sup]2[/sup] ) - sqrt( 5[sup]2[/sup] - ( x - 2.5 )[sup]2[/sup] )

Add the two areas together, then double it (since the above was only dealing with the upper semicircles). Unfortunately, my memory of the trig substitutions neccesary to integrate sqrt( a[sup]2[/sup] - x[sup]2[/sup] ) have been replaced by Buffy episode descriptions and video game cheat codes since college

Oh, and I’m assuming you wanted the larger crescent (on the left in the way I set up the graph). If you wanted the crescent with the outer edge formed by the **smaller** circle, find the areas under:

sqrt( 5[sup]2[/sup] - ( x - 2.5 )[sup]2[/sup] ) - sqrt( 5.25[sup]2[/sup] - x [sup]2[/sup] ) from [1.7625 , 5.25 ]

and

sqrt( 5[sup]2[/sup] - ( x - 2.5 )[sup]2[/sup] ) from [5.25 , 7.5]

On preview, I see **g8rguy** with his little namby-pamby explanation with x’s, y’s, and r’s, and it’s functionally the same as mine (including leaving the actual integration up to the student). But I typed all this out, damn it, I’m posting!