Question

The width of a circular ring is equal to the radius of the inner circle of the ring if the radii of the inner and outer circle are r and R then the area of the circular ring is
A. $2\pi {R^2}$
B. $2\pi {r^2}$
C. $3\pi {R^2}$
D. $3\pi {r^2}$

Answer 1

Hint: Draw the figure by using the conditions given in the question and use the standard formula of the area of a ring which is, $A = \pi ({R^2} - {r^2})$, here A is the area of the ring, R is the outer radius and r is the inner radius of the ring.

Complete step-by-step answer:

Given a ring of inner radius r and outer radius R.
According to the question, the width of the circular ring is equal to the inner radius r.
Therefore, outer radius $R = r + r = 2r$.
Now, area of the region between two concentric circles with radius of outer circle R, and inner circle r is given by-
$A = \pi ({R^2} - {r^2})$.
So, the area of the given ring$ = \pi ({R^2} - {r^2})$.
Put $R = 2r$ in the above equation of area of the ring, we get-
$
  A = \pi ({(2r)^2} - {r^2}) \\
  A = 4\pi {r^2} - \pi {r^2} \\
  A = 3\pi {r^2} \\
$
Hence, the area of the given circular ring is $3\pi {r^2}$.

So, the correct answer is “Option D”.

Note: For such a type of question, proceed step by step, as given in the question that a ring with inner radius r and outer radius R is there. Now as mentioned in the solution, the outer radius is 2 times the inner radius as the width of the ring is equal to the inner radius, this implies R = 2r, now using the standard equation of the area of a circular ring mentioned in the solution which is given by $A = \pi ({R^2} - {r^2})$, it is the area of the region between two concentric circles with radii R (outer radius) and r (inner radius). Then use the condition R = 2r. Then solve the equations to reach the solution.