Two circles are drawn such that the radius of the larger circle is 3 times the radius of the smaller...

1. TRANSLATE the problem information

• Given information:
  • Radius of larger circle = 3 × radius of smaller circle
  • Area of larger circle = 180π square centimeters
  • Need to find: Area of smaller circle

• Let \(\mathrm{r}\) = radius of smaller circle
• Then radius of larger circle = \(\mathrm{3r}\)

2. INFER the approach

• Since we know the area of the larger circle and need the area of the smaller circle, we can use the circle area formula to find the radius first
• Strategy: Use \(\mathrm{A = \pi r^2}\) for the larger circle to find \(\mathrm{r}\), then calculate area of smaller circle

3. APPLY the area formula to the larger circle

• Area of larger circle = \(\mathrm{\pi(radius)^2}\) = \(\mathrm{\pi(3r)^2}\) = \(\mathrm{\pi(9r^2)}\) = \(\mathrm{9\pi r^2}\)
• We know this equals 180π, so: \(\mathrm{9\pi r^2 = 180\pi}\)

4. SIMPLIFY to find \(\mathrm{r^2}\)

• Divide both sides by π: \(\mathrm{9r^2 = 180}\)
• Divide both sides by 9: \(\mathrm{r^2 = 20}\)

5. Calculate the area of the smaller circle

• Area of smaller circle = \(\mathrm{\pi r^2 = \pi(20) = 20\pi}\)

Answer: B) \(\mathrm{20\pi}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly set up the radius relationship, thinking "3 times larger" means the larger radius is \(\mathrm{r + 3}\) instead of \(\mathrm{3r}\).

This leads to incorrect area calculations and typically results in confusion when trying to match their answer to the given choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{9\pi r^2 = 180\pi}\) but make algebraic errors, such as forgetting to divide by π first or incorrectly calculating 180 ÷ 9.

Common mistakes include getting \(\mathrm{r^2 = 60}\) (forgetting to divide by π) or making arithmetic errors, which may lead them to select Choice D (\(\mathrm{60\pi}\)).

The Bottom Line:

This problem tests whether students can correctly translate proportional relationships into algebraic expressions and systematically work through the area formula. The key insight is recognizing that when a radius is scaled by factor \(\mathrm{k}\), the area scales by \(\mathrm{k^2}\).