A square and a circle have the same perimeter. If the radius of the circle is 3, what is the...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
A square and a circle have the same perimeter. If the radius of the circle is \(3\), what is the length of one side of the square?
\(\frac{3\pi}{4}\)
\(3\)
\(\frac{3\pi}{2}\)
\(3\sqrt{\pi}\)
1. TRANSLATE the problem information
- Given information:
- Square and circle have the same perimeter
- Circle's radius = 3
- Need to find: side length of the square
2. INFER the solution strategy
- To compare perimeters, we need both measurements in mathematical form
- Start with the circle since we know its radius
- Then use the equality condition to find the square's side length
3. Find the circle's circumference
- Using \(\mathrm{C = 2\pi r}\) with \(\mathrm{r = 3}\):
- \(\mathrm{C = 2\pi(3) = 6\pi}\)
4. TRANSLATE the equal perimeter condition
- Square's perimeter = \(\mathrm{4s}\) (where \(\mathrm{s}\) = side length)
- Circle's perimeter = \(\mathrm{6\pi}\)
- Equal perimeters means: \(\mathrm{4s = 6\pi}\)
5. SIMPLIFY to find the side length
- \(\mathrm{4s = 6\pi}\)
- \(\mathrm{s = \frac{6\pi}{4}}\)
- \(\mathrm{s = \frac{3\pi}{2}}\)
Answer: (C) \(\mathrm{\frac{3\pi}{2}}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students may confuse radius with diameter, thinking the circle's circumference is \(\mathrm{2\pi(6) = 12\pi}\) instead of \(\mathrm{2\pi(3) = 6\pi}\).
This leads to the equation \(\mathrm{4s = 12\pi}\), giving \(\mathrm{s = 3\pi}\), which doesn't match any answer choice. This causes confusion and random guessing.
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly set up \(\mathrm{4s = 6\pi}\) but make an arithmetic error when reducing the fraction, perhaps getting \(\mathrm{s = 6\pi}\) or \(\mathrm{s = \frac{3\pi}{4}}\).
This may lead them to select Choice (A) (\(\mathrm{\frac{3\pi}{4}}\)) or causes them to get stuck and guess.
The Bottom Line:
This problem tests whether students can accurately translate word relationships into mathematical equations and perform fraction simplification with \(\mathrm{\pi}\) terms. The key insight is recognizing that "same perimeter" creates a direct equality between two different geometric formulas.
\(\frac{3\pi}{4}\)
\(3\)
\(\frac{3\pi}{2}\)
\(3\sqrt{\pi}\)