A circular garden has a sector that subtends a central angle of 45^circ. The area of this sector is 8pi...

1. TRANSLATE the problem information

• Given information:
  • Sector has central angle of \(45°\)
  • Sector area = \(8\pi\) square meters
  • Need to find circumference of entire garden

2. INFER the approach

• To find circumference, we need the radius first
• Use the sector area formula to find radius, then apply circumference formula

3. TRANSLATE into sector area equation

• Sector area formula: \(\mathrm{A} = (\theta/360°) \times \pi\mathrm{r}^2\)
• Our equation: \(8\pi = (45°/360°) \times \pi\mathrm{r}^2\)

4. SIMPLIFY the equation step by step

• Simplify the fraction: \(45°/360° = 1/8\)
• Substitute: \(8\pi = (1/8) \times \pi\mathrm{r}^2\)
• Multiply both sides by 8: \(64\pi = \pi\mathrm{r}^2\)
• Cancel \(\pi\) from both sides: \(64 = \mathrm{r}^2\)
• Take square root: \(\mathrm{r} = 8\) meters

5. INFER the final step needed

• Now that we have radius, use circumference formula
• \(\mathrm{C} = 2\pi\mathrm{r} = 2\pi(8) = 16\pi\) meters

Answer: C. \(16\pi\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse sector area with total circle area and try to use the full circle area formula \(\mathrm{A} = \pi\mathrm{r}^2\) instead of the sector area formula \(\mathrm{A} = (\theta/360°) \times \pi\mathrm{r}^2\).

Using \(\mathrm{A} = \pi\mathrm{r}^2\), they would get: \(8\pi = \pi\mathrm{r}^2\), so \(\mathrm{r}^2 = 8\), giving \(\mathrm{r} = 2\sqrt{2}\). Then \(\mathrm{C} = 2\pi\mathrm{r} = 4\pi\sqrt{2}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(8\pi = (45°/360°) \times \pi\mathrm{r}^2\) but make algebraic errors when solving. They might forget to multiply by 8 when isolating \(\pi\mathrm{r}^2\), or make arithmetic errors with the fraction \(45/360\), leading to an incorrect radius and wrong circumference.

This may lead them to select Choice A (\(8\pi\)) or Choice B (\(12\pi\)) depending on the specific calculation error.

The Bottom Line:

This problem requires connecting two different circle formulas through a common variable (radius). Students must resist the urge to use the simpler full-circle formulas and instead work systematically through the sector relationship.