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
A regular hexagon can be divided into 6 equal equilateral triangles.
So each one has an area of 64√3
let's look at one of those.
Area of a triangle = (1/2)absinØ, where a and b are sides with Ø as their contained angle.
If each side is x, then
(1/2)x^2 sin60° = 64√3
(1/2) x^2 (√3/2) = 64√3
x^2 = 256
x = 16
take it from here
A. 4 B.2
Two triangles area similar the sides of one area 4,6,and 7cm,the short est side of the ather is 10cm calculate the lengths of the other two sides of this triangle
Answer
a. Each side of a regular hexagon can be found by using the formula:
Side length = √(Area / ((3√3) / 2))
Plugging in the given area of 384√3, we have:
Side length = √(384√3 / ((3√3) / 2))
Simplifying, we get:
Side length = √(768 / 3)
Side length = √256
Side length = 16
So, each side of the hexagon is 16 units long.
b. The radius of a regular hexagon can be found by using the formula:
Radius = Side length / √3
Plugging in the side length of 16, we have:
Radius = 16 / √3
Approximating the value, we get:
Radius ≈ 9.24
c. The apothem of a regular hexagon can be found by using the formula:
Apothem = Side length / 2
Plugging in the side length of 16, we have:
Apothem = 16 / 2
Apothem = 8
So, the apothem of the hexagon is 8 units long.
To find the answers to these questions, we need to use the formulas and properties of regular hexagons.
a. The formula to find the area of a regular hexagon is:
Area = (3√3/2) * s^2
where s is the length of each side of the hexagon.
Given that the area is 384√3, we can substitute this into the formula and solve for s:
384√3 = (3√3/2) * s^2
Divide both sides by (3√3/2):
(384√3) / (3√3/2) = s^2
Simplify the right side:
(384√3) * (2 / (3√3)) = s^2
(256√3) / √3 = s^2
Simplify further:
256 = s^2
Take the square root of both sides:
√256 = s
Therefore, the length of each side of the hexagon is 16.
b. The radius of a regular hexagon can be found using the following formula:
Radius = s / 2
where s is the length of each side of the hexagon.
Substitute s = 16 into the formula:
Radius = 16 / 2
Radius = 8
Therefore, the radius of the hexagon is 8.
c. The apothem of a regular hexagon can be found using the following formula:
Apothem = s * √3 / 2
where s is the length of each side of the hexagon.
Substitute s = 16 into the formula:
Apothem = 16 * √3 / 2
Simplify:
Apothem = 8√3
Therefore, the apothem of the hexagon is 8√3.