Question:A circle in the coordinate plane has an area of 36pi square units. What is the circumference of this circle?66pi1212pi36

1. TRANSLATE the problem information

• Given information:
  • Area = \(36\pi\) square units
  • Need to find circumference

2. INFER the approach

• To find circumference, we need the radius first
• We can get radius from the given area using \(\mathrm{A = \pi r^2}\)

3. SIMPLIFY to find the radius

• Set up the area equation: \(\pi\mathrm{r^2} = 36\pi\)
• Divide both sides by \(\pi\): \(\mathrm{r^2 = 36}\)
• Take the square root: \(\mathrm{r = 6}\)
• (We take the positive root since radius must be positive)

4. TRANSLATE and calculate circumference

• Use circumference formula: \(\mathrm{C = 2\pi r}\)
• Substitute \(\mathrm{r = 6}\): \(\mathrm{C = 2\pi(6) = 12\pi}\)

Answer: D (\(12\pi\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize they need to find radius first, and instead try to directly relate area to circumference without an intermediate step.

Some students might attempt to use the given area value directly in circumference-related calculations, leading to confusion and potentially selecting Choice C (12) by dropping the \(\pi\).

Second Most Common Error:

Poor SIMPLIFY execution: Students might make algebraic errors when solving \(36\pi = \pi\mathrm{r^2}\), such as forgetting to divide out the \(\pi\) or making square root mistakes.

This could lead them to get \(\mathrm{r = 36}\) instead of \(\mathrm{r = 6}\), which would give circumference = \(2\pi(36) = 72\pi\). Since this isn't an option, this leads to confusion and guessing.

The Bottom Line:

This problem tests the fundamental understanding that you often need to work backwards from one circle measurement to find the radius, then use that radius to calculate other measurements. The key insight is recognizing the two-step process rather than trying to find a direct relationship.