A regular hexagon is to be cut from a sheet of diameter d.the width across corner is
I will assume that by "a sheet of diameter d"
you mean a circle.
Sketch 6 diameters, with a 60° angle between them.
sketch your hexagon and label the points A, B, C, D, E, and F
Not sure which line you want by "across corner"
But, ...
If you look at triangle ACE, you have a right-angled triangle,
since CE is a diameter. It is easy to see that this is also
a 30-60-90 triangle with ratios of sides of 1 : √3 : 2
So, depending which "across corner" side you want:
AE/d = 1/2, ---> AE = d/2
AC/d = √3/2 ---> AC = √3 d/2
To find the width across the corner of a regular hexagon cut from a sheet of diameter d, we need to use some mathematical formulas.
1. The formula for the distance across the flats of a regular hexagon is given by:
Distance across flats = 2 * s,
where s is the length of each side of the hexagon.
2. The formula for the distance across the corners is given by:
Distance across corners = 2 * r,
where r is the distance from the center of the hexagon to any corner (also known as the circumradius).
To find the value of r, we can use the formula:
r = d / 2,
where d is the diameter of the sheet.
With this information, we can calculate the width across the corner of the hexagon.
Step-by-step calculation:
1. Calculate the length of each side of the hexagon (s):
Since a hexagon has six equal sides, we can use the formula for the circumference (C) of a circle to get the value of s:
C = 2 * π * r,
where π is a mathematical constant (approximately 3.14159) and r is the radius of the circle (half of the diameter).
Here, since we have the diameter (d), we can substitute it into the formula:
r = d / 2,
C = 2 * π * (d / 2),
s = C / 6.
2. Calculate the distance across the corners of the hexagon:
Distance across corners = 2 * r = 2 * (d / 2) = d.
Therefore, the width across the corner of the hexagon is equal to the diameter of the sheet, which is d.
To determine the width across the corners of a regular hexagon, you can use some basic trigonometric principles.
First, let's define the width across corners as "W".
In a regular hexagon, all sides are congruent, and each angle measures 120 degrees. The diagonal (width across corners) divides the hexagon into two congruent, equilateral triangles.
Now, let's consider one of these equilateral triangles. The length of each side of the equilateral triangle is equal to the radius of the hexagon (which is half of the diameter, d). We can denote this length as "r".
Using trigonometry, we can find the length of the diagonal of the equilateral triangle, which is also the width across corners of the hexagon.
In an equilateral triangle, the length of the diagonal (D) can be found using the formula:
D = 2 * r * sin(60 degrees)
Since the diagonal of each equilateral triangle is also the width across corners of the hexagon, we can substitute "W" for "D" in the equation:
W = 2 * r * sin(60 degrees)
Now, substituting the value of "r" (which is equal to half of the diameter, d/2) in the equation, we get:
W = 2 * (d/2) * sin(60 degrees)
W = d * sin(60 degrees)
To compute this value, sin(60 degrees) can be calculated using a scientific calculator or reference table. It equals √3/2.
Finally, substituting this value back into the equation:
W = d * (√3/2)
W = (√3/2) * d
Therefore, the width across corners of a regular hexagon cut from a sheet with a diameter of "d" is (√3/2) multiplied by the diameter "d".