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Calculating angles on an X leg table

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Blog entry by Rick_M posted 11-18-2017 06:06 AM 579 reads 0 times favorited 4 comments Add to Favorites Watch

I’m putting this here so the next time someone asks this question, I can find the information quickly. I have not tried these. Feel free to comment.

Calculating the angles of an X leg table is deceptively complicated because the width of the legs or braces changes the angles. Here are some alternatives to calculating the angle:
  • Tack the legs or braces together in an X and arrange them to the right angle.
  • Make a mock up from cardboard or plywood
  • Draw it full size on paper or plywood
  • Use a drawing program like Sketchup, etc.

But if you want to try and calculate the answer, maybe this will help…


Let B be the angle (initially unknown) the leg makes with the floor.
Let T be the thickness of the leg stock.
Let W be the width of outer sides of the X.
Let H be the height of the top of the X from the floor (i.e. the height of the under side of the table top.)
And let F be the width of the foot of the X legs (i.e. the length of the diagonal cut across the leg stock.)

Then sin(B) = T/F (looking at the triangle made by the foot and the thickness)
and
sin(B) = H / sqrt((W – F)^2 + H^2) (looking at the triangle the whole leg makes with the floor.)

Setting the two right sides equal and solving for F, (yeah, it s a mess) will let you compute the angle B (arcsin(T/F)). Then the angle you want is 180 – 2B. (Remember to convert arcsin(T/F) to degrees first.)

Just for reference (and to check my typing skills)

F =( (2T^2W) +/- sqrt((4T^4W^2)-4(T^2-H^2)) )/ 2(T^2-H^2)

(I told you it was a mess.)

- GeBeWubya

JBay found this calculator, but I haven’t tried it.


The math is over my head, Sketchup, it only takes a few minutes.
I did find this, (if it works) There is a calculator on the right side you can try.
http://jsfiddle.net/h1xda7qm/

- jbay

-- http://thewoodknack.blogspot.com/



4 comments so far

View Lazyman's profile

Lazyman

1506 posts in 1226 days


#1 posted 11-18-2017 11:35 PM

I could not quite get my mind wrapped around GeBeWubya’s formulas. Not sure if the formula is wrong or I just don’t understand his nomenclature? I did a little more searching and stumbled upon a link that helped me understand the math well enough to see how to calculate the angle. Here it is if you are interested:
https://math.stackexchange.com/questions/1757495/the-picnic-table-puzzler?answertab=active#tab-top
and you can play with the variables here:
https://www.geogebra.org/m/TasKWtAH
Actually a pretty cool tool. Note that this solution assumes you know the length of the board before cutting off the corners. I extrapolated from that to use the height and width as a variable instead of the length of the board.

So I took these formulas and put them into an XL spreadsheet and it seems to work. The resulting angle is slightly different than the calculator JBay’s linked to but that may be due to the rounding and truncation differences in the Javascript compared to none XL. I double checked my calculations on a calculator and got the same answer as my spreadsheet. It also turns out slightly different than the Sketchup model I was using to try to figure out the math on my own (yeah right) but that may be due to slight errors in some of the dimensions as I drew it.

Below I pasted the formulas from my spreadsheet. You should be able to copy & paste these into excel, modify the references to T, W, H, G & L to point to the correct cells and calculate the angle.

T =1.5 Enter Thickness of the board
W =72 Enter Width of tabletop
H =28 Enter Height above the ground
G = sqrt(H^2+W^2) Calculated Length of the corner to corner diagonal calced using width and height
L = sqrt(G^2-T^2) Length or board before cutting the corners off calced using diagonal and board thickness

Angle =degrees(pi()/2-atan(L/T)+asin(H/sqrt(L^2+H^2)))

-- Nathan, TX -- Hire the lazy man. He may not do as much work but that's because he will find a better way.

View Rick_M's profile

Rick_M

10640 posts in 2219 days


#2 posted 11-19-2017 05:23 AM

Thanks Nathan. Nice find.

-- http://thewoodknack.blogspot.com/

View Rick_M's profile

Rick_M

10640 posts in 2219 days


#3 posted 11-19-2017 05:31 AM

Totally off the subject but I still use those squares, awl, and knife you made me from the swap sometime back. The squares are holding up well. I love the different sizes.

-- http://thewoodknack.blogspot.com/

View Lazyman's profile

Lazyman

1506 posts in 1226 days


#4 posted 11-19-2017 01:08 PM

This formula actually makes sense to me so NOW I think of it as an “easy way” to compute the angles. Plus it also yields the lengths so provides another double check against the manual layout methods. Of course for the application that we started this conversation, you have to reverse the height and width because it assumes that the bevels are against the underside of the table top and on the ground rather than against the sides of the legs. I didn’t include the computation for the angles where the X intersects but those are easy to figure out using complementary angles and isosceles triangles.


Totally off the subject but I still use those squares, awl, and knife you made me from the swap sometime back. The squares are holding up well. I love the different sizes.

- Rick_M


Thanks Rick. I am glad they are holding up and finding use in your shop. I actually was about to ask how they are holding up as I was thinking of making a few more so want to make adjustments. I was a little worried about whether I hardened the awl well enough. I drop tested a prototype on concrete until I got it hard enough that the tip didn’t blunt but using a toaster oven to temper it was not the most exacting tool for the process. I also found a good plan in Woodsmith magazine for making a square out of aluminum and brass which might be fun to try.

Even further off topic…I browsed through your woodknack blog a few months ago and found your template for measuring RPM. This was handy for testing the variable speed motor I added to my lathe. Thanks.

-- Nathan, TX -- Hire the lazy man. He may not do as much work but that's because he will find a better way.

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