Area between two circles

Question

Let there be two circles of radius a and b such that b is greater than a. The area between the two circles would be : π(b²-a²). But why isn't it π(b-a)². Obviously I am looking for a more mature answer supported with a rigorous proof if any. Thanks.

Answer 1

You cannot take the area between the two circles to be $\pi (b-a)^2$ since doing so will give you the area of a circle with radius $b-a$ which is not what you are looking for. Imagine calculating the area of the figure left over after removing the smaller circle from the bigger one and then on the other side calculating area of a circle with a radius equal the the thickness of the leftover figure mentioned before. both areas will appear to diverge as the two circles get bigger with their differences in radii getting smaller which clearly says that they both will not be equal.

Answer 2

The area of the big circle is $\pi b^2$.

The area of the small circle is $\pi a^2$.

Now subtract the two : $\pi b^2-\pi a^2 = \pi (b^2-a^2)$.

Note that $\pi (b-a)^2=\pi(b^2-2ab+a^2)$ which is not the correct answer.

Answer 3

The radius does not get smaller, that is what you say when you use $b-a$. But the area of the circle is given by $\pi a^2$ and $\pi b^2$ and the difference gives:

$$\pi b^2-\pi a^2=\pi(b^2-a^2)$$