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What is circle inside a circle (like that of a pipe's upper part) called and what is its area???

It is called concentric circle. which area do you want---the inner concentric circle or the area between the two concentric circles?
The region between two concentric circles is called annulus



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Area = area of outer circle - area of inner circle
        =π(R² - r²)

See the attachment.
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I think its a ring 
area of it is=  \pi (R {2}  -r {2} )
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