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# Geometry

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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.

Apr 28, 2024

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We are given two facts about a regular hexagon:

Perimeter (p): We know the total length of all sides of the hexagon is p units.

Area (A): We are told the area of the hexagon is A=23​ square units.

We want to find the side length of the hexagon.

Here's how we can approach this problem:

Relate Perimeter and Side Length: A regular hexagon has six sides, all with the same length. Let's call this side length "s" (units). The perimeter (p) can be expressed as the total length of all sides:
p = 6s (equation 1)

Relate Area and Side Length: There's a formula for the area of a regular hexagon based on its side length "s":
Area (A) = (3√3 * s^2) / 2 (equation 2)

Solve for Side Length: We are given A = 3/2 and want to find s. We can use equation 2 and substitute the given value of A:

(3/2) = (3√3 * s^2) / 2

Solve for s:

Multiply both sides by 2: 3 = 3√3 * s^2

Divide both sides by 3√3: s^2 = 1 / √3

Take the square root of both sides (remembering there might be positive and negative square roots): s = ± (1 / √3)^(1/2)

Since side length cannot be negative, we take the positive square root: s = 1/√3

Simplify s: Recall that the radical (√) symbol represents the square root. We can simplify 1/√3 by rationalizing the denominator, multiplying both top and bottom by √3:

s = (1/√3) * (√3/√3) s = √3 / 3

Therefore, the side length of the regular hexagon is √3 / 3 units.

Apr 28, 2024