What is the length of one side of a square that has the same area as a circle with radius...

1. TRANSLATE the problem information

• Given information:
  • Circle has radius 2
  • Square has the same area as this circle
  • Need to find: side length of the square

2. INFER the solution approach

• To solve this, I need to:
  • Find the area of the circle first
  • Set that equal to the area formula for a square
  • Solve for the square's side length

3. Calculate the circle's area

• Using \(\mathrm{A = πr^2}\) with \(\mathrm{r = 2}\):
• \(\mathrm{A = π(2)^2}\)
  \(\mathrm{= π(4)}\)
  \(\mathrm{= 4π}\)

4. TRANSLATE "same area" into an equation

• Square area formula: \(\mathrm{A = s^2}\) (where \(\mathrm{s}\) = side length)
• Since areas are equal: \(\mathrm{s^2 = 4π}\)

5. SIMPLIFY to find the side length

• Take square root of both sides: \(\mathrm{s = \sqrt{4π}}\)
• Use the property \(\mathrm{\sqrt{ab} = \sqrt{a} × \sqrt{b}}\):
• \(\mathrm{s = \sqrt{4} × \sqrt{π}}\)
  \(\mathrm{= 2\sqrt{π}}\)

Answer: C. \(\mathrm{2\sqrt{π}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly set up \(\mathrm{s^2 = 4π}\) but then make algebraic errors when simplifying \(\mathrm{\sqrt{4π}}\).

Common mistakes include:

• Writing \(\mathrm{\sqrt{4π}}\) as \(\mathrm{\sqrt{4} + \sqrt{π} = 2 + \sqrt{π}}\)
• Confusing it with \(\mathrm{4\sqrt{π}}\)
• Getting \(\mathrm{\sqrt{2π}}\) (perhaps by incorrectly "factoring out" something)

This may lead them to select Choice B (\(\mathrm{\sqrt{2π}}\)) or get confused and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might think the side length should simply equal the radius, missing that the problem asks for equal areas, not equal linear dimensions.

This intuitive but incorrect reasoning leads them to select Choice A (2) - just using the radius directly.

The Bottom Line:

This problem requires students to distinguish between equal areas versus equal linear measurements, then carefully manipulate square root expressions. The algebra with square roots is where most calculation errors occur.