The area of a regular hexagon is equal to the area of a square. If both figures lie in the same plane, what is the ratio of the perimeter of the regular hexagon to the perimeter of the square?

Introduction / Context:
This question compares a regular hexagon and a square which have exactly the same area. The goal is to relate their perimeters and express the result as a ratio of the perimeter of the regular hexagon to the perimeter of the square. This type of problem tests understanding of area formulas, how side lengths scale between different shapes, and how perimeters change when areas are matched between figures.

Given Data / Assumptions:

• The polygon is a regular hexagon with side length a.
• The square has side length s.
• Areas of the two figures are equal.
• Both figures are assumed to be in the same plane.
• We use standard exact formulas for areas and perimeters and then compute a numerical ratio.

Concept / Approach:
The key ideas are as follows:

• Area of a regular hexagon with side a is A_hex = (3 * sqrt(3) / 2) * a^2.
• Area of a square with side s is A_sq = s^2.
• Perimeter of the hexagon is P_hex = 6a.
• Perimeter of the square is P_sq = 4s.
• First equate the areas to relate a and s, then form the ratio P_hex / P_sq and evaluate it numerically.

Step-by-Step Solution:
Step 1: Write the area of the regular hexagon: A_hex = (3 * sqrt(3) / 2) * a^2.Step 2: Write the area of the square: A_sq = s^2.Step 3: Set the areas equal: (3 * sqrt(3) / 2) * a^2 = s^2.Step 4: Take square roots to relate sides: s = a * sqrt(3 * sqrt(3) / 2).Step 5: Perimeters are P_hex = 6a and P_sq = 4s.Step 6: Form the ratio P_hex / P_sq = (6a) / (4s) = 3a / (2s).Step 7: Substitute s from Step 4 into the ratio and simplify to get a numerical value approximately equal to 0.93.Step 8: Hence the ratio of the perimeter of the hexagon to the perimeter of the square is approximately 0.93 : 1.

Verification / Alternative check:
One numerical way to verify is to choose a convenient value for a, such as a = 1, compute the exact area of the hexagon, then solve for s from s^2 = A_hex.Next, compute P_hex = 6a and P_sq = 4s numerically with that value of s.The resulting quotient P_hex / P_sq is approximately 0.93, which confirms the earlier symbolic derivation.Trying a different starting value of a will give the same ratio, because both perimeters scale linearly with a.

Why Other Options Are Wrong:
Options such as approximately 1.07 : 1 or 1.20 : 1 suggest that the hexagon has a larger perimeter than the square for equal areas, which is not supported by the calculations.Values like approximately 1.50 : 1 and 0.80 : 1 deviate even more from the true ratio and do not match the perimeter relationship derived from the area equality.Only the option approximately 0.93 : 1 matches the correct numerical evaluation of the ratio.

Common Pitfalls:
A frequent mistake is to directly compare side lengths instead of areas when forming the ratio of perimeters.Another error is to assume that equal areas imply equal perimeters, which is not true for different shapes.Some learners may also misapply or forget the correct area formula for a regular hexagon, or they may incorrectly simplify the square root expression when solving for s.

Final Answer:
The ratio of the perimeter of the regular hexagon to the perimeter of the square is approximately 0.93 : 1.