Magic Formula for Drawing Octagons
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- Опубліковано 15 кві 2022
- #octagon #octagonratio #drawoctagon
The formula for drawing a beautiful octagon may seem magic, but it’s simple math using the .414 ratio!
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In other episodes, we’ve addressed various methods for drawing or cutting octagons.
“Create A Perfect Octagon Every Time!” is a fast and low-tech method to draw an octagon when you know the side-to-side dimension of the octagon you want. This approach only takes a ruler, pen, and compass. You can check that episode out at:
• Create A Perfect Octag...
“EZ Method to Cut Octagon Stock” is a great way to cut octagonal column-shaped pieces - and requires no calculations. This is great for cutting off the corners of turning stock. Check it out at:
• EZ Method to Cut Octag...
Use this website for definitions of different parts/segments of octagons - www.omnicalculator.com/math/o...
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Great video! I didn't know about the .414 factor!
Also to clarify, I believe the inside angles are 135°. Each side is 45° from the adjacent side, but the interior angle of each corner is 135°.
If you drew a triangle from two outside points to the center, the angle from the center would be 45°.
If you drew lines from one single point to each of the other points, the difference between each of those lines would be 22.5°. I'm not sure if this is the same 22.5° that you were referring to.
Anyway, you prompted me to think more deeply about octagons than I ever have before! 😂🤘
Klay, thanks for watching and taking time to explain your thoughts. Perhaps the obscurity comes from the way that I said it. If you take any of the side segments of an octagon, and then measure the angle that the next segment "heads off" in, that angle is 45°. This happens 8 times around the octagon, equaling 360°. That is what you are stating with each side being 45° to adjacent sides. You are correct in saying the interior angles are 135°.
As a maker/woodworker, I'm interested in the angle from side to (adjacent) side - 45°. When cutting pieces, I would need each end of each segment (side piece) to have ends on it that are 22.5°. That aligns with your calculation as well.
However, knowing the .414 factor, I can predetermine the length of each segment if I know how "wide" I want the octagon to be.
Thanks for writing and all the observations! Best Regards, DFJ
Another great tip Jay. Always great information on your videos.
Glad to help!
Great info to have!
Good afternoon from Southeast South Dakota
Interesting
I have a 15 1/4" length by 15 1/4" width square to work with now if I take 15 1/4" x 0.414 = 6.3135. I want my octagon to be the width. I get the 6 but when I try to find the simplest fraction form for .3135 it gave me 3135/10000
= 627/2000 which this is where I need simplest fraction form so it will 6 ( ?)/(? )" tell me if I am right I turned 5/16 fraction into a decimal for I came up with 0.31. So would each side of the octagon all 8 sizes equal to the length to be cut at 6 5/16" fraction on my 2x4's? Would I make
my markings on the inside or outside for my miter cuts? I have never tried to make a octagon shape out of wood. I want to try to its very intriguing to learn something new .
Bradley - my apologies for the very slow reply. I didn't see it until today - it was inside one of the comment areas in UA-cam instead of where I am used to seeing them.
So, let's answer your questions. You want to make an octagon shape that measures 15-1/4" from side to side. I assume that's the outside edge of the octagon you are making, not the inside, correct? If so, multiply 15.25 x .414 to get 6.31. Then utilize a fraction to decimal chart to get your nearest fraction equivalent.
Which side you make the markings for the cuts depends if you want the dimensions we've worked out to be the inside or outside edge of the shape you are making. Does that help?
Best, DFJ
What if I was building a Navajo octagon hogan house ? Would it still work in foot & inches x .414 ??
Randall, indeed. Just determine the side to side measurement, but do it all as inches, not foot and inches. Take the result times .414, and that will give you the side segment length of each wall section. Let's say your hogan is going to be 16' from side to side. 16' in inches is 192". Take that times .414, that equals 79.49 inches, or right at 79-1/2" per each of the 8 sides. Does that help? Best, DFJ
@@DirtFarmerJay Yes it helps thanks a lot 👍🏾🙏🏽
Why dont you show were the number is coming from. Tan 22.5 is sin45/(1+cos45)= square root (2)-1= 0.4142135624
Square root of 2 - 1 is much easier to remember
Mari - I love it! My approach is for the majority of mere mortals that don't have the understanding of geometry that you possess. Wow - that is a great formula that shows the "why" of the .414 number. Thanks for sharing and raising my intelligence a bit. Best Regards, DFJ
You go into to much detail You should get to the point!
Thanks for stopping by. Another channel would probably more to your liking. Thanks for the feedback, though. Best, DFJ