Find the perimeter of a regular hexagon that has an area of 54 radical 3 units squared.
In wikipedia type hexagon
Area:
A=t^2*[3*sqroot(3)]/2
54*sqroot(3)=t^2*[3*sqroot(3)]/2
Divide booth sides with sqroot(3)
54=3*t^2/2
54*2=3t^2
108=3t^2 Divide booth sides with 3
108/3=t^2
t^2=36
t=6
Perimeter:
P=6*t=6*6=36
To find the perimeter of a regular hexagon, we need to first calculate the length of one side.
The formula for the area of a regular hexagon is:
Area = (3 * sqrt(3) * s^2) / 2
Given that the area is 54√3 units squared, we can solve this equation for s (the length of one side):
54√3 = (3 * sqrt(3) * s^2) / 2
Let's simplify this equation:
Multiply both sides by 2 to get rid of the fraction:
108√3 = 3 * sqrt(3) * s^2
Divide both sides by 3 * sqrt(3) to isolate s^2:
s^2 = (108√3) / (3 * sqrt(3))
s^2 = 36
Now, take the square root of both sides to find the length of one side:
s = √36
s = 6
Since all sides of a regular hexagon are equal in length, we know each side measures 6 units.
To find the perimeter, we multiply this length by the number of sides, which is 6:
Perimeter = number of sides * length of one side
Perimeter = 6 * 6
Perimeter = 36 units
Therefore, the perimeter of the regular hexagon with an area of 54√3 units squared is 36 units.
To find the perimeter of a regular hexagon with a given area, you can follow these steps:
1. Recall the formula for the area of a regular hexagon: A = (3√3/2) * s^2, where A is the area and s is the length of each side.
2. Set up the equation using the given area: 54√3 = (3√3/2) * s^2.
3. Simplify the equation: 2 * (54√3) = 3√3 * s^2.
4. Divide both sides of the equation by 3√3: (2 * 54√3) / (3√3) = s^2.
5. Simplify the expression: (108√3) / (3√3) = s^2.
6. Cancel out the common terms: 36 = s^2.
7. Take the square root of both sides to find the length of each side: √36 = s. Therefore, s = 6.
8. Since a regular hexagon has six equal sides, the perimeter of this hexagon is 6 * s = 6 * 6 = 36 units.
So, the perimeter of the regular hexagon with an area of 54√3 units squared is 36 units.