+1911 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.
+743 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.
