Calculate the area of the crescent

Assuming AD is the diameter of the smaller circle and C is the center of the larger circle.

If $CD = x$ then, $CE = 4+x$.

Note that angle DEA is a right triangle.

We have by the similarity of triangles EDC and ACE that

$\frac{x}{4+x} = \frac{4+x}{9+x}$

Solving gives $x = 16$. Thus the radius of larger circle is $25$. The radius of the smaller circle is $\frac{x + 9+x}{2} = 20.5$

Area of the crescent = $\pi ((25)^2 - (20.5)^2) = 204.75 \times \pi$

Here's how I'd do it:

Call $R$ the radius of the big circle and $r$ that of the small one. Now, observe that the surface of the crescent is just the difference between the surface of the big circle and the small one or

$$\pi R^2 - \pi r^2 = \pi (R-r)(R+r) \; .$$

Note how I've expressed this surface as a product of two quantities which I am now going to determine from the other data in the drawing. First the difference between the double radii is clearly 9cm:

$$2R-2r=9$$

We're halfway. Then the distance between the center of the small circle and point E is obviously $r$, but can alternatively be expressed with Pythagoras as

$$r^2 = (R-5)^2+(r-R+9)^2 \; .$$

Reordering and using what we already know about $R-r$:

$$r^2 - (R-5)^2= \left(\frac{9}{2}\right)^2 \; .$$

Again, using the factorizing trick

$$(r - R+5)(r+R-5)= \left(\frac{9}{2}\right)^2 \; .$$

Thus,

$$r+R= 5+2\left(\frac{9}{2}\right)^2 \; .$$

Combining everything, we get that the surface of the crescent is

$$\pi \frac{9}{2}\left(5+2\left(\frac{9}{2}\right)^2\right)$$

found the original post here: https://softwareengineering.stackexchange.com/questions/20927/what-is-your-favorite-whiteboard-interview-problem/28439#28439

and there is the solution in the comments:

the diameters differ by 9cm, so if the inner circle has radius r, the outer circle has radius r + 4.5. The area of the crescent is the difference in the areas of the circles: pi(r + 4.5)^2 - pi * r^2. All that's left is finding r. Define C as the point (0,0), then point E is at (0, r - 0.5) (because CE is 5cm less than the larger radius). The inner circle is shifted right 4.5cm, so its equation is (x - 4.5)^2 + y^2 = r^2. Plug in (x,y) = (0, r - 0.5) and solve for r.