Question

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

Answer 1

The side length of a regular hexagon with area
`$3$` is
`$√(2)$`. The formula for the area,
`\(A = (3√(3))/(2) \cdot s^2\),` yields
`\(s = √(2)\).`

The area $A$ of a regular hexagon with side length
`$s$` can be expressed in terms of its side length using the formula:

`\[A = (3√(3))/(2) \cdot s^2.\]`

In this case, you are given that
`$A = 3.$` Therefore, you can set up the equation:

`\[3 = (3√(3))/(2) \cdot s^2.\]`

To solve for the side length
`$s,$` you can rearrange the equation:

`\[s^2 = (2)/(√(3)).\]`

Now, take the square root of both sides to find the side length:

`\[s = \sqrt{(2)/(√(3))}.\]`

To rationalize the denominator, multiply the numerator and denominator by the conjugate of
`$√(3)$`:

`\[s = \sqrt{(2)/(√(3))} \cdot (√(3))/(√(3)) = \sqrt{(6)/(3)} = √(2).\]`

So, the side length of the regular hexagon is
`$√(2).$`