What is (find) the area of the ring between the circles.?

Question

Two concentric circles are given. A tangent to the inner circle forms a chord twelve inches long in the larger circle.

Answer 1

Say the radius of the larger circle is R and that of the smaller one, r. Draw a perpendicular from the center of the circles to the tangent point and from the center to the point the tangent intersects the larger circle.

You will see that there is a right triangle with one leg r, the other 6 inches and the hypotenuse R.
r**2 + 36 = R**2. This gives 36=R**2 - r**2
But the area required is Pi (R**2 - r**2) = 36 Pi