Magic Formula for Drawing Octagons

Hey, Dirt Farmer Jay here from DirtFarmerJay.com. Being able to draw an octagon accurately, easily, and to a predetermined size is a really cool thing. Well, we've given you a tip already how to do this in the past, but I'm giving you another mathematical way today that I'm sure you're gonna find really handy. Stay tuned. Hi, Dirt Farmer Jay here from DirtFarmerJay.com. Thanks for joining me today. If you're a maker, a woodworker, being able to draw octagonal shapes is really important. In a previous episode, and you can check that out the link right above me here, or in the description below, we showed you how to draw an octagon like this one down here. Now, if you look careful, you can see that I have some of my draft marks here. And the other method I've shown you is essentially drawing of a square, marking the center, determining the width from the center of the corner, and using that distance to mark off all eight points of an octagon. Before we go any further, let me define an octagon. And you're probably going, come on, everybody knows an octagon. But let's give you a little bit more formal definition. Of course, the most common octagonal shape that we see is the stop sign. And so we're so used to the shape, but perhaps have never thought about elements that put it together. So if you look down on the side here, you can see from side to side is a dimension. And these are parallel. Each of these are parallel from the one across from it. There are eight sides, all equidistant, all equal lengths. And each of these angles right here are 45 degrees. And there's eight of them. If you take eight times 45, you get that 360 degrees that takes you back to the beginning. That also means that each of the half angles that comprise this full angle are 22 and a half degrees. Now for you woodworkers, and I've done this quite a bit here, this is really handy information. Just setting up a sliding, a sled on your table saw, or doing a really accurate job of setting up your miter saw to cut 22 and a half degree angles at predetermined lengths. What we did in this one here, this is a 13. I'll have you take a look here. If you look across from side to side here, and I'll do it so you can see it from you, this is a 13 inch wide octagon. So here we have a 13 inch, and it doesn't matter which diameter I go from or which angle dimension is what I should say. You're still gonna get it. Let me adjust that over there. There we go. We have a 13 inch side to side measurement. Well, let's show you another really cool factor that's used to predetermine the sides. All right, so if you look down here, you can see that we have that 13 inch side to side, and that was a predetermined amount. I wanted an octagon that was 13 inches. But did you know what that one piece of information, side to side width, I can tell you without actually laying this out, what that length is right there, that length that's gonna get used all the way around on those 22 and a half degree angles? Absolutely, I can. And here is the secret. It is the 0.414 factor. So all you're going to do is take the width of the octagon side to side, or top to bottom, or angle to angle. It's all the same measurement. And you're gonna multiply that times 0.414. Then you can simply use a fraction to decimal table to come out with the closest to a 64th, that say, if you're getting at a 128, you're probably getting a little too far, but that's up to you, how far you wanna take it accuracy-wise. And you're just gonna multiply that 0.414 times the width, and then get it to the closest fraction. So when I take 13 inches, and I multiply it by, in this case, 0.414, I end up with five and three-eighths inches. Now the decimal is 5.38, da-da-da, which just happens to be about the same number as three-eighths. That's not how it usually works out. But the closest 64th was five and three-eighths. Well, let's just take a look and see if that really works. Take a look down here. Again, I'm gonna reverse the ruler. We're just double-checking here. I'm going across here, and you can see that we have a 13-inch length on the ruler of a side-to-side measurement. Now, let's just choose this one because it's easiest for you to look at. By looking at that factor, I should have five and three-eighths and five and three-eighths from point to point right here. Well, let's see how we did. I'm gonna take my zero, put it there, and look at that. We are right at five and three-eighths. And I knew that was going to the dimension before I ever drew it. So this is really handy when you want to figure out sides, or let's suppose you're doing a mirror or something like that has an angled top of long sides. You can figure all that out before you actually draw it and then confirm it with a pattern or drawing like this. Now, just to test this out to see if it really worked, we did something else, but we're gonna show you that in just a moment. Before we do that, may I ask you a favor? If you like the way that we provide material to you, it's really helpful to us if you like the video. And when you do, better yet, subscribe, and better yet, ring the bell, because that way, approximately every Saturday, you'll get a notification of the latest and greatest video that we put out about the home, the garden, homeowner skills, the kitchen, vehicle, just all sorts of things where you can just do it yourself. We appreciate the support. All right, so what we did to verify this, we visited the stop sign at the end of our street and measured it. And as you can see, I just hooked the tape over the top. Why? You don't have to go side to side. You can go any parallel surface, diagonal, top to bottom, side to side, and it was easier to hook the tape over the top. So we just measured top to bottom, standard stop sign in the United States, 30 inches. And then we went ahead and measured the diagonal, but I could tell you where it's going to be, because when you take 30 inches and you take it times 0.414, you're going to get about 12.42 inches. And the closest one, or closest 16th, it's a little less than 12 and 7 16ths. So less than a half, more than 3 8ths. So it's in that area right there. And as you can see, when we measure it, there it is. That ratio is entirely scalable up and down and works great. You know, another thing that is a skill that is really helpful when you're building is being able to draw beautiful, fair, smooth arcs. Check out this video over here where we show you two of four methods, and there's another follow-on that shows two more methods to drawing beautiful arcs. And then check out this other video where YouTube has gotten into your mind and knows where you live and is putting some things there for you that'll be of interest to you. Until the next time, this is Dirt Farmer Jay from DirtFarmerJay.com.