the area of a regular hexagon is given as 384√3
a. how long is each side of a hexagon
b. find the radius of a hexagon
c. find the apothem of a hexagon
Your hexagon consists of 6 equal equilateral triangles, of course the angle
would be 60°
If the side of the hexagon is x,
then the area of each triangle = (1/2)(x)(x)sin60° = √3 / 4 x^2
you have 6 of them, so the area of the hexagon = 6√3/4x^2
= (3/2)√3 x^2
then (3/2)√3 x^2 = 384√3
x^2 = 256
x = 16
I bet you can finish it from here
To solve the given problems, we need to use the formulas for the area, side length, radius, and apothem of a regular hexagon. Let's start with the formula for calculating the area of a regular hexagon:
Area of a regular hexagon = 3 × (√3/2) × side^2
Given that the area of the hexagon is 384√3, we can set up the equation:
384√3 = 3 × (√3/2) × side^2
Now we can solve for each part of the problem:
a. Finding the length of each side of a hexagon:
Divide both sides of the equation by 3 × (√3/2) to isolate side^2 and find its square root:
side^2 = (384√3) / (3 × (√3/2))
side^2 = 128√3
Therefore, the length of each side of the hexagon is √(128√3).
b. Finding the radius of a hexagon:
The radius of a regular hexagon is the distance from the center to a vertex, and it is also the distance from the center to the midpoint of a side. We can find the radius using the formula:
Radius = side / √3
Substituting the value of the side length we found from part a, we get:
Radius = √(128√3) / √3
Radius = 8√3
Hence, the radius of the hexagon is 8√3.
c. Finding the apothem of a hexagon:
The apothem of a regular hexagon is the distance from the center to a side, which is also the radius of the circle inscribed within the hexagon. We have already found the radius in part b, so the apothem of the hexagon is also 8√3.
To find the length of each side of a regular hexagon, you can use the formula:
Side length (s) = sqrt(area / ((3 * sqrt(3))/2))
a. Substitute the given area, which is 384√3:
s = sqrt(384√3 / ((3 * sqrt(3))/2))
s = sqrt(384√3 / (3 * sqrt(3)/2))
s = sqrt(384√3 * (2/ (3 * sqrt(3))))
s = sqrt(768 / 3)
s = sqrt(256)
Therefore, the length of each side of the hexagon is 16.
To find the radius (r) of the hexagon, you can use the formula:
Radius (r) = side length (s) / 2
b. Substitute the side length we found:
r = 16 / 2
r = 8
Therefore, the radius of the hexagon is 8.
To find the apothem (a) of the hexagon, you can use the formula:
Apothem (a) = side length (s) * sqrt(3) / 2
c. Substitute the side length we found:
a = 16 * sqrt(3) / 2
a = 8 * sqrt(3)
Therefore, the apothem of the hexagon is 8√3.