Geometry Forum - Problem of the Week
Solutions - Hexagonal Window, Nov. 14-18, 1994
Annie says:Many people were willing to wade into this everyday problem containing "messy" numbers and practical considerations to be addressed. Most found the lengths of the sides using the known ratios of sides in 30-60-90 triangles, although the Burnaby folks brought the Law of Sines into play. Some paid careful attention to the cutting, realizing that the pieces which are cut off the ends affect the length of the piece you need to start with (13 inches is the length of a side but the piece of board from which those are cut is longer).
We wrote back to a number of students who initially imagined the cut pieces as trapezoids instead of parallelograms. The biggest problem was that many ignored the comment that the studs would form two sides and without drawing a picture it was easy to think that 6 sides of 13 inches each would be required. In fact, this won't even work. Sean did a nice job in thinking through the cutting process and realizing that one form could be used for all four boards.
It was interesting to see no attention given to the engineering considerations, for instance, would the joint at the bottom of the window be strong enough to hold together without additional bracing across the bottom? Is there enough wood for that?
Eric Wahl
Cut the wood into 4 segments 15 inches long. For each of the 4 segments, cut across the 3.5 inch surface to make a parallelogram 13 inches long with sides 4 inches and angles of 60 degrees and 120 degrees. These 4 pieces can then be put together with a 13 inch length on each stud to form a regular hexagon. Since the studs had to be 2 of the sides, the hexagon has a vertex at the top and the bottom. Using the width between the studs, I divided the hexagon into a square 22.5 in. wide and 4 30-60-90 triangles with 11.25 in. 6.495 in. and 12.99 in.Sean Nichols
A regular hexagon can be divided up into 6 equilateral triangles, all joining at the center of the hexagon, the length of their sides being equal to the radius of the circle circumscribed about the hexagon. Therefore, this length is also equal to the length of one side of the hexagon itself. Starting with the side formed by the vertical wall stud on the left of the proposed window, we can label the sides a, b, c... in a clockwise motion about the hexagon, so we have:
| b/\c |
|/ \|
a | |d
| |
|\ /|
| f\/e |
We can take the perpendicular distance between a and d to be
22.5" (GIVEN), and so half this distance (the hexagon's apothem
to a) is equal to 11.25". Now let's look at the triangle with
side a:
|\
| \
| \ x
| \
a |____\
| z /
| /
| / y
| /
|/
z = 11.25" (shown above). Define the acute angle formed by sides
x and y to be @, and the acute angle between y and z to be þ.
Since the z is perpendicular to a, we can use simple trigonometry
to show that cos(þ) = z/y. þ = @/2 and @ is defined to be 60 (the
angle between any two sides in an equilateral triangle = 60
degrees). Therefore, angle þ has a value of 30 degrees. So, since
cos(þ) = z/y, y = z/cos(þ) = 11.25"/0.866025403... = ~12.99038106..."
(sqrt of 168.75, to be exact - kinda hard to measure with a
ruler!). As I have already shown, this is equal to one side of
the hexagon.So the size needed for a piece of wood for one side would be 12.99..." long. Therefore, our pieces of wood will have the following dimensions:
b
__
/ /|
/ /þ| c
/ / |
a/ / /
/ / /
/ / /
/__/@ /
| x| /
| | /
|__|/
a = 12.99..."
b = 3.5" (GIVEN) - (Assuming thickness of wall = 3.5") (**Sean,
better check your thinking here for the length of c, it is NOT
given**) [c = 1.5" (GIVEN)]
@ = 120 degrees
þ= 60 degrees
x = 90 degrees (Right angle).
This one form can be made four times, which will suffice, and
all fit from the 8' board given. (sides b and e - see above -
can be made in this way, and sides c and f can be the same
shape "turned around" (flipped 180ƒ horizontally). Sides a and d
are formed out of the original studs.)
Jennifer Hinkel & Colleen Brophy
1) We made a scale drawing where .5cm=1inch.2) Knowing that the measure of each angle of a regular hexagon is 120 degrees, we drew a perpendicular at the vertex so that there was a 30 degree angle on each side (to form a 180 degree angle).
3) To find the exact measure of each side we used the 30-60-90 theorem based on the Pythagorean theorem.
4) We must cut 4 equal pieces of wood 13 inches long in which a parallelogram is for other measuring 120 degrees and 2 acute angles opposite each other with measures of 60 degrees.
Susan Garges
Briana and Brandt should design the frame for the hexagonal window so that the angles and sides are all equal. Each interior angle would be 120 degrees. Each exterior angle would be 60 degrees. You can now find the measure of the sides using the Pythagorean theorem. Because you know 2 of the angles (60 and 90) and one side (22.5/2=11.25) there is only one way you can close off the triangle. (11.25)^2 + (6.5)^2 = c^2. So c is approximately 13 inches.Kathleen Graham
The parallel wall studs supply the two completely vertical sides of the frame. To calculate the length that each side must be, one has to divide the distance, 22.5 inches, between the wall studs by 2. This answer is 11.25 inches. Draw a perpendicular line from the midpoint between the two wall studs to one of the studs. This line is 11.25 inches. Then draw a line from the midpoint to the end of the same stud. By doing so, a 30-60-90 triangle has been formed. Since the tangent is equal to the opposite side over the adjacent side, the tan 60 = 11.25/adjacent side. Therefore, the adjacent side is 6.50 inches. 6.50 inches is half the length of the side of the stud. Multiply 6.50 inches by 2 and the answer is 13 in.Previous page || Next problem || Previous problem || Table of Contents || Forum Home Page