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A regular hexagon has a perimeter of $p$ (in length units) and an area of $A$ (in square units).  If $A = \frac{3}{2},$ then find the side length of the hexagon.

 Dec 18, 2023
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Here's how to find the side length of the hexagon:

 

Formula for area of a regular hexagon: The area of a regular hexagon can be calculated using the formula:

A = (3√3 / 4) * s^2

 

where A is the area and s is the side length.

 

Substituting known value and solving for s: We are given that A = 3/2 square units. Plugging this into the formula:

 

3/2 = (3√3 / 4) * s^2

3/2 * 4/3√3 = s^2

2/√3 = s^2

 

Taking the square root of both sides (remembering both positive and negative solutions):

 

s = ± √(2/√3)

 

Simplifying the radical:

 

s = ± √(2√3 / 3)

 

Therefore, the possible side lengths of the hexagon are:

 

s1 = + √(2√3 / 3) ≈ 1.051 units (approximately)

s2 = - √(2√3 / 3) ≈ -1.051 units (negative value not applicable)

 

Since the side length of a polygon cannot be negative, the valid side length of the hexagon is approximately 1.051 units.

 

Note: This answer provides the solution in both exact and approximate forms. You can use whichever form is more appropriate for your specific context.

 Dec 18, 2023

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