By what percentage should the radius of the circle be reduced to reduce its area by 19%.

Let the area of the circle be S1, then the area of the reduced circle, by condition, is S2 = S1 – S1 * 0.19 = 0.81 * S1.

The area of the first circle is: S1 = n * R12. (1).

The area of the reduced arm will be equal to: S2 = n * R2 ^ 2 = 0.81 * S1.

Then S1 = (n * R2 ^ 2) / 0.81. (2).

Equate Equations 1 and 2.

n * R1 ^ 2 = (n * R2 ^ 2) / 0.81.

R2 ^ 2 = R1 ^ 2 * 0.81 = R1 ^ 2 * 0.9 ^ 2.

R2 = R1 * 0.9.

R1 – 100%.

R1 * 0.9 – X%.

X% = 100 * 0.9 = 90%.

100% – 90% = 10%.

Answer: The radius needs to be reduced by 10%.