A chord to the larger circle is tangent to the inner smaller

A chord to the larger circle is tangent to the inner smaller circle. If the length of the chord is 16 cm, find the area of the ring or annulus (the region between the two circles). Note the figure is not drawn to scale.

Solution

Let R be the radius of the bigger circle and r the smaller circle

Then area of the ring of annulus = Area of big circle - Area of small circle

= pi (R²-r²)

Since AB is the tangent distance of AB from C = r

Since chord of big circle is 16 cm, consider the right angled triangle formed by CDB where D is the touching point.

CD =r, CB =R Hence R²-r² = (16/2)^2 =64 sq cm. (By Pythagorean theorem)

Hence Area of the ring = 64 pi cm^2