The math behind segmented wood rings

so i have an upcoming project where i need to make a wood ring like the one shown here these are not my wood rings uh credit to the window company they belong to but you can see that they are joined with individual pieces of wood and then milled out to make a circle i started to do the math on how many pieces of wood you need and how wide those pieces need to be and it turned out that the math was a little trickier than i thought so i wanted to share it with you in case you have a similar project so let's say this is the circle that you want to make and you have to decide how many pieces of wood it's going to take to form the template for making the circle you could start by just using four pieces of wood and joining them together into a square but you can see that given the particular board widths here and the side of the circle that you're not going to completely cover it you're going to have these parts of the circle that are missing so then you might look at well what if i just add more sides the same circle with a pentagon so i've got five pieces of wood and this circle actually is completely enclosed and once you have the minimum number of sides you can actually add as many sides as you want so if you wanted to do it with a hexagon that would also work you could go up to seven sides eight sides once you hit the minimum you can keep increasing the number of sides but it's sort of unnecessary and increases the complexity to the work so the question i wanted to answer was what is the minimum number of sides that you need to create a wood circle so let's start with a circle and i'll show you the math that i ended up with so let's assume we're going to make a circle with an inner radius r that's the inner radius and the circle is going to have a thickness of t and the here are the boards we're dealing with and they have a width w so if this was a two by four w would be three and a half inches the first constraint is that the board width w needs to be greater than the thickness of the circle that's sort of intuitive if you have a narrower board you you're not going to be able to encompass the circle given that board width and given the radius and the thickness of the circle the first thing you do is calculate the number of sides of the polygon that you're going to create whether it's a pentagon or a hexagon or whatever and this is the formula so it's 180 divided by the inverse cosine of this fraction and this is written out to be in degrees and that's going to give you some number n of the number of sides you need now that number is going to be a fraction 4.3 or whatever and we want to deal in a regular polygon so you just round it up to the next highest number if you get 4.3 you go up to five you're dealing with a five-sided figure a pentagon okay so let's assume that those calculations lead to a pentagon being the shape that we need to make that pentagon we need to know two more things we need to know the miter angle the angle that we're going to cut each of these boards and we need to know the side length of each of the sides of the pentagon so figuring out the minor angle is straightforward it's just 180 divided by the number of sides figuring out the ideal side length is a little trickier as you make the side a little bit smaller you push this outer edge of the circle towards the edge of the board and as you make the side length longer you push the inner edge of the circle closer to these vertices so what you'd like to do ideally is pick a side length that centers this circle as best as possible in the structure which is to say you have the same margin of error against this outer board as you do against these vertices here is the formula for that side length it's a bit complicated but essentially it's two times the tangent of the miter angle times this whole expression and again that will be the side length that best positions the circle within the polygon that you've built so just to recap first you calculate the number of sides that you're going to need according to this formula it's going to be some decimal and you're going to round that up to the next highest whole number at that point you can calculate the miter angle just by dividing 180 by the number of sides and then finally you need the side length which is given again by this formula so one of the big design choices you have when building these structures is what with board to use and the wider the board that you use the fewer sides that you're going to need but the more wood you're going to waste so i thought it would be useful to set out the calculations for determining the amount of material that you're using when you make the ring so to do that we need the area of the polygon which is given by this formula it's the width of the board times this expression the side length minus the width of the board times the tangent of the miter angle that is the area of each one of these trapezoids and then we have n number of those trapezoids and we want to compare that to the area of the ring which is given by pi r squared of the outer edge minus pi r squared of the inner edge and if we take the ratio of those two numbers we'll get a percentage indicating how much of the material we've actually used so i created an excel spreadsheet where i input the formulas that i just walked through and it accepts as inputs the inner radius of the ring that we want to build the thickness of the ring the width of the boards that we want to build with and it outputs the number of sides the miter angle the side length and a percentage of the material that will be used so let's type in a couple examples say we want to build a ring with an 18 inch inner radius three inches thick and we're going to use three and a half inch wide boards in that case we would need a 14 sided figure here's the miter angle here's the side length and we'd be using 85 percent of the wood let's type in the same ring but with wider boards five and a half inches in that case you can see that the number of sides that we'll need goes way down from 14 to 6 but the percentage of material we're using also goes down so there is a trade-off between build complexity which is to say how many pieces we need to cut and join and the efficiency with the amount of material that we're using if we wanted to try to get that efficiency really high then we can use a board width that's very close to the thickness of the ring that we're trying to build so if i did a five inch thick ring and use 5.1 inch boards we would have to do a ton of sides 36 but we're using 98 of the material so let's enter one that i can build in my garage as a test say a six inch inner radius with a 2.5 inch thickness and i'm going to use two by fours which are three and a half inches wide so i'm going to need to build a hexagon and my side lengths are going to have to be just shy of 10 inches and my efficiency is going to be not so great about two-thirds of the wood will be used so let's go build that so here i have my hexagon six sided figure 30 degree miters i glued it up let it dry [Music] according to our table we needed the sides to be just under 10 inches and that's what they are we're building a six inch radius that is 12 inch diameter circle so we need the longest point inside this hexagon which is from vertex to vertex to be less than 12 inches which it is at just over 11 and three quarters inches i input 2.5 for the thickness of the ring so the diameter of the outside circle is going to be 6 plus 6 plus 2.5 plus 2.5 or 17 inches total so we need the shortest distance from the outside of this hexagon to be more than 17 inches and if we've done our side calculations right it should be about quarter inch more because that's the margin of error we had on the inside and we had picked our side length to have the same margin of error on the inside as on the outside and that's what it is at 17 and a quarter inches so we know that we can fit inside this wood structure the ring that we wanted and all we would have to do is draw the ring and use whatever tool a bandsaw or a router to cut it out i'm not going to do that because i don't need a ring of this size but for my next project i need a larger ring so i'll show you in that project

Channel: Project ReaDIY

Views: 23,137

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Id: IKuZ63DARQY

Length: 10min 10sec (610 seconds)

Published: Thu Jul 15 2021