The small circle and the large circle have the same center O. Chord AB is tangent to the smaller circle at C. Prove that the area between the large and small circle is the same as the area of the circle that has AB as a diameter

Answer 1

If the two circles have radii r and R, then the area between the circles is

π(R^2-r^2)

If a circle has AB as a diameter, then BC is its radius, since
AB = AC + BC

BC^2 + r^2 = R^2
so,

BC^2 = R^2-r^2

That means the area of the circle with BC as its radius (AB as its diameter) is

π BC^2 = π(R^2-r^2)