The sphere in the figure is of radius 15 cm. How do I find the surface area of the shaded part of the sphere in figure 1? Please help!

As you see the inner triangle has sides $20cm$ and $15cm$ and is right angled, so the third side (that from O to the top of the triangle) must be $25cm$ (it's pythagorean). Also as the radius of the circle is $15cm$ you would have that the height of the triangle is $40cm$.
This means that the the width of the triangle $20cm$ up from the base is half of the base (as it's half-way to the top).
Then you use the formula for the area of a trapezoid. It becomes $A = {30+60\over 2}20 cm^2$
The formula you need for the surface area of the shaded portion of the sphere is $A=2\pi rh$. You have $r=15$ and $h=20$, so $A=600\pi\,cm^2$.