Question

The radius of a circle is so increased that its circumference increases by 5%. The area of the circle then increases by:

• A. 12.5%
• B. 10.25%
• C. 10.5%
• D. 11.25%

Hint:

Circumference is proportional to radius, but area is proportional to the square of radius: square the 1.05 growth factor.

Answer 1

The Correct Option is B

Instead of working with percentage multipliers, pick an easy number for the radius and track the actual values.

1. 12.5%: this figure only appears if you add the circumference rise to itself plus a bit more, not from squaring the radius change.
2. 10.25%: this matches the exact value found by squaring a 5% radius increase.
3. 10.5%: this looks like doubling 5% and adding a small rounding, but it skips the square term.
4. 11.25%: this comes from an arithmetic slip in expanding the square, not the true value.

Let the original radius be $r = 100$. Since circumference is proportional to radius, a 5% rise in circumference makes the new radius $105$. Old area is proportional to $100^2 = 10000$, and new area is proportional to $105^2 = 11025$. The increase is $11025 - 10000 = 1025$, which is $\frac{1025}{10000} \times 100 = 10.25\%$ of the old area.

Let's summarize:

• Radius 100 to 105 is a clean 5% rise.
• Area goes from 10000 to 11025 units, a 10.25% rise.

This confirms option B, 10.25%, using plain numbers instead of algebra.