SPOILER AHOY
It gets even more intuitive.
Drawing the lines connecting the centre of the smaller circle with the middle and either end of the chord, we create a right triangle whose sides are 25/2m, r1, r2. Applying the Pythagorean Theorem: (25/2)^2 = r2^2 - r1^2
The required area is: pir2^2 - pir1^2 = pi(r2^2 - r1^2) = pi(25/2)^2