each leg of an octagon is 48" what is the diameter?
Formula for long diameter (vertex to opposite vertex),
d = (2 * side length * sin 67.5)/(sin 45)
Formula for short diameter (one side to opposite side)
d = ((2 * side length)/(sqrt(2))) + side length
I'll do short diameter,
d = (2 * 48)/(sqrt2)) + 48
d = (96)/(1.4142) + 48
d = 67.8829 + 48
d = 115.8829
I'm not sure if this is the way you were taught.
You can make use of the fact that if you draw a short diameter in the octagon, two 45-45-90 triangles are formed (with a rectangle between them).
The short diameter would then be, x + 48 + x.
x = one of the equal legs of the 45-45-90 triangle. The side length of 48 is the hypotenuse of each of the right triangles.
Using the properties of the sides being in the proportion of x : x : x(sqrt(2)),
you can find x (since you know the hypotenuse is 48).
To find the long diameter, another right triangle is formed.
The 'short diameter' found above is one leg, 48 is the other leg and the hypotenuse is the 'long diameter'.
Since you know two legs, you can use the Pythagorean theorem to find the long diameter.
To find the diameter of an octagon given the length of each leg, we first need to know the formula for calculating the diameter of a regular octagon.
A regular octagon is a polygon with eight sides of equal length. Let's denote the length of each leg as "s". In a regular octagon, the distance from one vertex to the opposite vertex (passing through the center) is known as the diameter (d).
The formula to calculate the diameter of a regular octagon is:
d = s / (2 * sin(π/8))
In this formula, "sin" represents the sine function, and "π" is a mathematical constant, approximately equal to 3.14159.
Now, let's substitute the given value of the leg length, which is 48 inches, into the formula to determine the diameter:
d = 48 / (2 * sin(π/8))
To evaluate this expression, let's calculate sin(π/8):
sin(π/8) ≈ 0.38268
Now, substitute this value back into the formula:
d = 48 / (2 * 0.38268)
Simplifying further:
d ≈ 48 / 0.76536
d ≈ 62.83185 inches
Therefore, the approximate diameter of the octagon is 62.83185 inches.