16-Sided Polygon

[55fireflite]

Greetings,

I’ll start with my question: If I created a 16-sided polygon, with all sides and angles equal, that measures 13.5 inches across, how long will each side be?

Here why I’m asking:

I volunteer at a historic site that interprets life in Pennsylvania circa 1700. I’m working on woodworking project that has presented me with this geometry problem. Our blacksmiths are using a 21st century five-gallon plastic water bucket to quench their workpieces, which looks decidedly out of place in a 17th century blacksmith’s shop.

My task is to build a wood “bucket” (with no bottom) to slip over the plastic one, concealing it from view. I decided to use 16 staves, forming a 16-sided polygon. 360 degrees ÷ 16 = 22.5 degrees for each angle. Divide that by two, and each edge of each board needs to be beveled at 11.25 degrees. Using reclaimed white oak from the site, I milled the boards flat, cut the width oversize, and beveled one edge. Now I just need to know how wide the inside surface needs to be, and I can set the fence on table saw to cut the other edge.

I tried doing a full size layout, drawing a series of 8 overlapping squares, but I couldn’t get it accurate enough to get a precise measurement.

Thanks,
Tom

 

[Dr.Peterson]

See equations (1), (3), and (5) here: http://mathworld.wolfram.com/RegularPolygon.html

Your "distance across" is either 2r or 2R (depending on whether you mean side-to-side, or corner-to-corner), and one side is a.

Assuming it is to be 13.5 inches across the inside from the middle of one stave to the opposite one, you want r = 13.5/2 = 6.75, so (using degrees rather than radians) a = 2r tan(180/n) = 13.5 tan(180/16) = 2.685 inches.

 

[Steven G]

360 degrees ÷ 16 = 22.5 degrees for each angle

Your math is correct, but where did the 360 degrees come from? Did you draw the polygon? Please do so.

 

[55fireflite]

Thank you both for your responses.
Dr. Peterson,
Sorry I wasn’t specific about that point. You are correct that my 13.5 inch dimension is from the center of one stave to the center of the opposite stave, i.e., the narrowest distance. A 5 gallon bucket measures just under 12 inches across. Add a bit for the wire handle and a bit of wiggle room, and I came up with 13.5 inches.

11 ÷ 16 = 0.6875, so 2 11/16 inches it is.

Jomo,
Here is a snapshot of my full-size rendering.





I drew this to help me figure out both the angles and the length of the sides.

I remembered the concept about 360 degrees while making my drawing. I figured I could create my polygon by drawing a series of squares. First I drew a 13.5 inch circle to represent the clearance I needed. Then I drew a square around that, touching the circle at the center of each side. Those tangents represented the inside surface of my staves.

It took a minute of head scratching to figure out how to draw the next square. Then I realized all I had to do was draw two lines connecting the opposite corners of the first square, which located the center of both the square and circle, creating four 90 degree angles. Then I draw two more lines through the center, splitting those angles in half. Draw four more tangent lines intersecting those lines at 90 degrees, and I had an octogon. Repeat the process two more times, and I had my 16-sided polygon.

When I realized I could locate the corners of the second square by drawing more lines through the center of the circle, it hit me. EUREKA! I didn’t need to measure the resulting angles on the drawing itself. Draw any number of lines through the center of a circle, and the angles between those lines will always add up to 360 degrees. I would have 16 radii an equal distance apart, so 22.5 degrees would be the angle between the staves.

I always did well in math, but I was a Poly Sci major, and I haven’t used the cosine button on a calculator in 35 years, so that’s as far as I was going to get on my own. Because of the limitations of my slightly wonky drawing, I only knew the inside width sides should be somewhere between 2 1/4 and 2 7/8. Not good enough.

Again, thank you very much for your assistance. Charter Day is next Sunday (celebrating the day William Penn got his Charter from Charles II), and I should have it ready to go by then. The staff will be thrilled.