A circle is inscribed in a square. The radius of the circle is 5sqrt(3) inches. What is the side length,...

1. TRANSLATE the problem information

• Given information:
  • Circle is inscribed in a square
  • Radius of circle = \(5\sqrt{3}\) inches
  • Need to find: side length of the square

2. INFER the key geometric relationship

• When a circle is inscribed in a square, the circle touches all four sides of the square
• This means the diameter of the circle exactly equals the side length of the square
• Strategy: Find diameter, which will be our answer

3. SIMPLIFY to find the diameter

• \(\mathrm{Diameter} = 2 \times \mathrm{radius}\)
• \(\mathrm{Diameter} = 2 \times 5\sqrt{3}\)
• \(\mathrm{Diameter} = 10\sqrt{3}\) inches

4. Apply the relationship

• Since diameter = side length of square
• Side length = \(10\sqrt{3}\) inches

Answer: D (\(10\sqrt{3}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that "inscribed" means the diameter equals the side length. They might think the radius equals the side length instead.

Students confuse the geometric relationship and use radius directly as the side length, leading them to select Choice B (\(5\sqrt{3}\)).

Second Most Common Error:

Conceptual confusion about inscribed circles: Students might visualize the problem incorrectly, thinking the circle somehow relates to the diagonal or corner of the square rather than touching the sides.

This confusion about the geometric setup leads to guessing or trying to use the Pythagorean theorem inappropriately, causing them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students can visualize the inscribed circle geometry and understand that "inscribed" specifically means touching all sides, creating the diameter-to-side-length relationship that makes this problem solvable in one step.