A circle with circumference 16pi inches is inscribed in a square. What is the length, in inches, of the diagonal...

1. TRANSLATE the problem information

• Given information:
  • Circle has circumference \(\mathrm{16π}\) inches
  • Circle is inscribed in a square
  • Need to find diagonal of the square

2. INFER the key geometric relationship

• When a circle is inscribed in a square, it touches all four sides of the square
• This means the circle's diameter exactly equals the square's side length
• This connection is crucial - without it, you can't relate the circle's measurements to the square's dimensions

3. SIMPLIFY to find the radius

• Use circumference formula: \(\mathrm{C = 2πr}\)
• Substitute: \(\mathrm{16π = 2πr}\)
• Divide both sides by \(\mathrm{2π}\): \(\mathrm{r = 8}\) inches

4. INFER the square's side length

• Since diameter = \(\mathrm{2r = 2(8) = 16}\) inches
• And diameter = side length of square
• Therefore: side length = 16 inches

5. SIMPLIFY to find the diagonal

• Use diagonal formula for squares: \(\mathrm{diagonal = side × √2}\)
• Diagonal = \(\mathrm{16√2}\) inches

Answer: C) \(\mathrm{16√2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what "inscribed in a square" means geometrically. Students might think the circle just sits inside the square somewhere, rather than touching all four sides. Without this key relationship, they can't connect the circle's measurements to the square's dimensions and get stuck trying to figure out how to proceed. This leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion: Mixing up radius and diameter when determining the square's side length. Some students correctly find \(\mathrm{r = 8}\), but then use 8 (the radius) as the side length instead of 16 (the diameter). This leads them to calculate diagonal = \(\mathrm{8√2}\), causing them to select Choice A (\(\mathrm{8√2}\)).

The Bottom Line:

This problem tests whether students can visualize the geometric relationship between inscribed shapes and apply that insight to connect different measurements. The calculations themselves are straightforward once you know what equals what.