A regular hexagon has an area of 96sqrt3. What is the length of each side of the hexagon?
The area of a polygon of n sides with an apothem of a and side c is
A=nac/2
For a hexagon,
n=6, apothem=c*sqrt(3)/2
so
Area=6*c*(sqrt(3)/2)*c/2
=3c²sqrt(3)
In the given case,
96sqrt(3) = 3c²sqrt(3)
Solve for c (length of side).
To find the length of each side of a regular hexagon, we can use the formula for the area of a regular hexagon:
Area = (3 * √3 * s^2) / 2
where "s" represents the length of each side.
In this case, the area of the hexagon is given as 96√3. So we can set up the equation as follows:
96√3 = (3 * √3 * s^2) / 2
To solve for "s", we can first simplify the equation by canceling out the common terms of √3:
96 = (3 * s^2) / 2
Multiplying both sides of the equation by 2 to get rid of the fraction:
192 = 3 * s^2
Dividing both sides by 3:
64 = s^2
Taking the square root of both sides:
s = √64
s = 8
Therefore, the length of each side of the hexagon is 8.