I am having a hard time determining what formula I need to solve the following: On a hexagonal window the height is 1.3m. Find the width.Do I use the pythagreom theorem (but I have only one known number) or do I use the formula 6(1/2as)? I'm confused. The book says the answer is 2.6/square root of 3 or 1.5m. How was this obtained?
Draw a regular hexagon. It consists of six equilateral triangles. Each has a "height" of 0.65m (measured between two opposite sides). The "width" of the hexagon, measured between opposite corners, is 2R, where R is the radius of the circumscribed circle, which is 0.65/sin60 = 0.75056 m
Double that for the width: 1.5011 m
Round off to 1.5.
thank you. I knew the equilateral triangles played a part, but I was confused because only one measurement was given.
To find the width of a hexagonal window given the height, we can use the formula for the distance between two opposite corners of a regular hexagon.
The formula for the distance between two opposite corners of a regular hexagon is given by:
d = s * √3,
where d is the distance between two opposite corners, and s is the length of one side of the hexagon.
In this case, we know the height of the hexagonal window is 1.3m. We want to find the width, which is the distance between two opposite corners. We can assume that the window is regular, meaning all sides and angles are equal.
Since we only have information about the height, we need to relate the height to the side length of the hexagon. To do this, we can draw a perpendicular line from one of the top corners to the base of the hexagon, creating two right triangles.
Using this approach, we have a right triangle with the height (1.3m) as one side and half the width (w/2) as the other side. The hypotenuse of this right triangle is the side length of the hexagon.
Applying the Pythagorean theorem to this right triangle, we have:
(1.3)^2 = (w/2)^2 + s^2.
Since we have two unknowns, w (width) and s (side length), we need an additional equation to solve for both. This is where we use the formula for the distance between opposite corners of a regular hexagon.
From the formula, we know that the distance between opposite corners (d) is equal to s * √3.
Since we now have two equations,
(1.3)^2 = (w/2)^2 + s^2, and
d = s * √3,
we can substitute the second equation into the first, which gives us:
(1.3)^2 = (w/2)^2 + (d/√3)^2.
We can solve this equation for the width, w.
However, it seems like the book has provided an alternative solution. The book mentions the answer is 2.6/√3 or 1.5m. To derive this value, we need to understand how to simplify the expression.
If we divide both the numerator and denominator of 2.6 by √3, we get:
2.6/√3 = (2.6/√3) * (√3/√3) = (2.6√3)/(√3 * √3) = (2.6√3)/(√9) = (2.6√3)/3.
So, the book's given answer of 2.6/√3 is equivalent to (2.6√3)/3.
Therefore, the width of the hexagonal window is either approximately 2.6/√3 or 1.5m, as stated by the book.