Can't calculate the angles needed for cuts - anyone good at mathmatics?

[Akuma]

Hey everybody.

I am not exactly sure, whether this topic should be in the woodworking section. I want to create a custom speaker shelf top which looks slightly like the top of the Sega Astro City cab. To give you an idea, I googled for pics. Here is one:



Copyright notice: Source here: http://case.oncle-tom.net/2006/07/23/sega-astro-city/

I am talking about the top in shape of a trapezoid (is it called that way?) which holds the marquee and the speakers.

For this thing, I made a little sketch. The top are the four pieces in a 3D view. Beneath this, is what it should look like. Being bad at math for my whole life, my problem is that I just can't figure out, how to calculate the angles (green) which I need to cut off the blue and red marked areas to make it fit.

First, I hope that you understand what I am talking about and what I want to achieve.

Second, maybe someone has an idea - I've been thinking about this for hours now and my brain is kinda fried at the moment : /

[bfauska]

It's not exactly what you are asking for (figuring that out with the measurements supplied is proving to be quite difficult) but it is probably really close to what you want.  If it doesn't work I could figure it out closer to what you are asking with a little more info, but see if this works first.

[Neverending Project]

As bfauska said, it is difficult to determine the important dimensions fro mwhat you have given. Normally you use the height, top and bottom dimensions to fit your trapezoid into a specific area. It is unclear if you need to keep the sides at 180mm, or if there is another dimension that is more important to keep.

That being said, if bfauska's drawing doesn't quite work for you, attached is a spreadsheet that may help you get the dimensions you need. You can change the top and bottom lengths and thicknesses or the overall height, and it will calculate the angles and the side length. I hope this helps.

[Edit]Updated File[/Edit]

[Xiaou2]

 Just my .02$


  I personally do this by running some scrap wood thru the tablesaw.

 Make small cuts,  as you can then increase the cut angle as needed.

 Place the boards together to mock up the placement...  then test
fit the part.   If the angle is not quite as expected,  try making
another pass with a slight angle change.

 Once the proper angle has been produced...   Grab the Good wood,
and run both parts thru the tablesaw.

 (one upside up, and other upside down)


 Since you have two angles to worry about on each piece,  then after
both the first initial cuts have been made on the good wood...  Then
make an additional set of pieces on scrap wood.    Make enough
so that you dont run out if making too many mistakes.

 Follow the same procedure.   Use a marker to draw the ideal angle
first... then cut slightly behind it  - just in case.   Mock test it.  Then
adjust blade angle as needed.     Once fit,  use the good wood,
making sure the angles are facing the proper directions.

 Sure, you might be able to get it down mathmatically... but,  once you
try to cut it on the tablesaw..  how will you be sure that the angles
and material removal depths will be correct?    Rare to get that kind
of accuracy.

 Ive build a few advanced angle setups using the method described above.

 (btw - you may wish to start out with a cardboard mockup to get an
idea of how you want it mounted, and the look and angles desired.   I
imagine mounting will be tricky because you are trying to mount to
plastic.   If the wood is too heavy, it could easily crack the plastic
shell.   Unless you have a system of bracing internally, which is
quite strong)

 Best of Luck

 
 

[leapinlew]

 Just my .02$


  I personally do this by running some scrap wood thru the tablesaw.

 Make small cuts,  as you can then increase the cut angle as needed.

This is good advice. Me though, I make small cuts, get frustrated, screw it in and fix it with bondo.

Actually... I just don't do many cabs with too many complicated angles.

[ScottS]

Holy crap! Has the entire world forgotten trigonometry?

I think you need to use the inverse trig functions (e.g. arccosine) to figure out the angle given the lengths of two sides...

[Akuma]

Erm, just to let all you guys hear something from me - at the moment I am working on the control panel and after this is done, I will focus on this marquee/ speaker top - so I really appreciate everyone's answers, I just decided to take care of the cp first.

[shardian]

Lay out your top and bottom pieces on some paper and trace them. Now lay your side pieces according to the paper. Now make marks one the pieces where they intersect the outlines of the top and bottom piece. Voila, you have your cut lines. and no trig necessary.

[leapinlew]

Holy crap! Has the entire world forgotten trigonometry?

I think you need to use the inverse trig functions (e.g. arccosine) to figure out the angle given the lengths of two sides...

Seems there is at least 1 person who remembers trigonometry.


Showoff... 

[bfauska]

Holy crap! Has the entire world forgotten trigonometry?

I think you need to use the inverse trig functions (e.g. arccosine) to figure out the angle given the lengths of two sides...

Scott, with the info given, we don't have the lengths of the sides.  I am familiar with basic trig and the answer to the question as posted is beyond that.

[ScottS]

Scott, with the info given, we don't have the lengths of the sides.

I haven't actually done the math, but it looks to me like we have the dimensions of all the pieces and a diagram of how he intends to put them together. What more do we need?

[bfauska]

To do it with as basic of a trig calculation as you are suggesting, we need the dimensions of the finished shape, not the dimensions of the pieces.  The numbers I gave are within a few thousandths of a MM, but in order to get the ACTUAL answer it would take several more steps, the problem is that you have to figure out several little triangles at the end of each piece in order to figure out the bigger shape.  Since I can't put puzzles down until I've totally frustrated myself or solved them I'm going to give this some more thought, but it definitely isn't as easy as just figuring out the angles in a simple trapezoid.

[bfauska]

   OK, so apparently total frustration isn't that difficult of an achievement for me.  I give up on finding the actual numbers.  What I do know is that the problem is some very tightly woven together trig and algebra.  The only way to figure out the angles is to know the dimensions of the finished outer sides (or inner sides, but you only need one or the other), in order to find those you need to know the angles, there are formulas for figuring out all the parts you need, but I am fairly sure that they would all have to end up fit into the sides of a deep dark algebraic equation.  Since there is only one way to accomplish what is drawn there must be a mathematical way to figure it out, but it's screwy enough that I've lost interest and hope.


Realistically though, I bet that the dimensions provided are not the actual requirements of the design, since the sizes of the parts shown do so little to demonstrate the actual finished size of the assembly, I think that there may be a mis-communication happening. 

If I were to measure something out in the world to copy it, I wouldn't have the dimensions of the parts I need, I would have the dimensions of the shape I want.  In the drawing I used in my first answer that is how I used those dimensions.  The odd part about this particular situation is that if you built what was shown with the parts that were shown only two places on that finished product would have any of the original dimensions; the bottom edges of both the top piece and bottom piece would be the lengths called out in the original problem.  Measuring the finished product would never give you the lengths called out for the sides in the original drawing, either you would get the measurement from the bottom corner to the top corner, or you would get the measurement of the part after it was cut down as shown in the assembly portion of the drawing.  Essentially what I am saying is 

Hope that all helps or something, or even makes the littlest bit of sense to somebody other than myself, although it probably won't, as things rarely do.    

[ScottS]

I think you're making this more complex than it needs to be. Let's start with a simple case, as illustrated by the following poorly drawn diagram:



Angle T1 can be found using:

T1 = arccosine((B-A)/2C)

Once you know T1, you also know T2:

T2 = 180 - T1

As you point out, the lengths A, B, and C need to be either all interior measurements or all exterior measurements. Because the interior and exterior triangles are similar, the angles don't change. Once you know the angles of the corners, you can easily figure the angles at which you need to saw the boards: if the angle looks like it's more than 90-degrees then it's T2, if it looks like it's less than 90-degrees then it's T1. Coming up with a formula that gives the exact length for the side boards is also pretty simple, I think. Something along the lines of:

Side Length = Total Outside Length - 2*(16mm / sine(T1))

I came up with this formula assuming the side pieces would be measured along their outside edge. Haven't done the math, but I suspect that the inside edge has the same length...

[scotthh]

I thought bfauska's original answer was totally adequate for wood working: getting a miter saw more accurate than a degree or two seems unlikely.

But if you're still not sure, download Neverending Project's Excel workbook. It enables you to put in a thickness for the wood. It has the trig functions in there. And it has a picture that helps keep everything in perspective. I highly recommend it! Yet almost no one has downloaded it.

[Akuma]

: ) Okay, building this badboy is no.1 on my "to do list" for tommorow - I will let everyone know how bfauska's calculations worked out

Cheers

[bfauska]

ScottS,

What you've drawn is the process to use if we had been supplied with the dimensions A, B, & C.  Those are the dimensions I believe Akuma is really interested in and what he's drawn is probably mis-communicating what he's looking for.  My point about the difficulty of calculating it was almost completely in an academic discussion, to calculate exactly the information that the original drawings requested would have been much more complex, almost definitely excessive, and a waste of time compared to getting the information in a different way.

Provided somebody knows the desired finished exterior dimensions or interior dimensions of their project the information posted by ScottS is correct.  Thank you Scott for posting it for us all to see.  Given the same known dimensions I could also do a drawing with no math involved and have the computer spit out the angles, the original problem was that we didn't have the finished dimensions of the part.

Oh, and yes I was making it more complex than it needs to be to build the project, but no more complex than the problem originally was.

Scotthh,
I'll have to give that workbook a look, it sounds interesting.  Thank you.

[Akuma]

ScottS,

What you've drawn is the process to use if we had been supplied with the dimensions A, B, & C.  Those are the dimensions I believe Akuma is really interested in and what he's drawn is probably mis-communicating what he's looking for.  My point about the difficulty of calculating it was almost completely in an academic discussion, to calculate exactly the information that the original drawings requested would have been much more complex, almost definitely excessive, and a waste of time compared to getting the information in a different way.

Provided somebody knows the desired finished exterior dimensions or interior dimensions of their project the information posted by ScottS is correct.  Thank you Scott for posting it for us all to see.  Given the same known dimensions I could also do a drawing with no math involved and have the computer spit out the angles, the original problem was that we didn't have the finished dimensions of the part.

Oh, and yes I was making it more complex than it needs to be to build the project, but no more complex than the problem originally was.

Scotthh,
I'll have to give that workbook a look, it sounds interesting.  Thank you.

Yes, I understand it's because I only bought the 4 pieces of wood in the given dimensions and not gave exact outside dimensions of the trapezoid I want to create (since I don't know them myself) : )

As i said, I will start work on it tommorow

[bfauska]

OK.  Now I get why those were the dimension you gave.  I assumed that you measured something and gave us the dimension that you figured would be the first cuts before the bevel went on.  It is also nice to know that you aren't really trying to match something exactly.

  and good luck on trying the build of it soon.

[Neverending Project]

OK. Now that everyone understands that the discussion (most of it anyway) is more academic than practical, here's a tip for the aforementioned spreadsheet. In Excel select Tools -> Goal Seek. In the Set Cell: box, you can enter "Ang1" or "B9" to  find a particular Angle. Then in the To Value: box, enter your desired angle - maybe 75. And in the By changing cell: box, enter the value you want it to change. If you enter "h" or "B5", it will tell you that by setting the height (h) to 203.41, you can get an angle of 75 degrees. As another example, by setting the height to 173.17 you can get the angled side to be exactly 150.

The formulas can also be modified to allow the user to type in the angle, and it would calculate the height or width. It all depends on what dimensions are fixed, and what you have to play with. Fitting this box inside an existing area would most likely necessitate that the overall height and width are fixed.

But now this is all just academic, no?

[bfauska]

Neverending P,

I just took a look at that Excel worksheet and it's pretty cool.  I think I skipped it before expecting it to be more like a spreadsheet that did the math of ScottS's solution which doesn't solve the original problem, and while this doesn't quite either it is much closer than I expected.  I will have to poke around in it later and see what you did.  Good clean geeky fun.

[TOK]

My best friend is a carpenter for a living. I don't think he even took Algebra 1, but he would figure out how to make that piece in a few minutes just using a tape measure.
I'd ask him how he'd do it, but he doesn't even have a computer. 

[Xiaou2]

Again,   Unless you have some super accurate table saw,  your math
wont mean too much.

 How will you know your table saws angle is exactly 132 degrees for
example?   

 And, are you accounting for the blade width removal in your calculation?
Which side will it remove from... and or will it be centered removal?

 What if your saws guide is slightly off? 

 There are many issues.   mostly to do with material removal of the
blade thickness, and the less that adequate angle measurement on
most tablesaws.

 This is why its much easier and accurate to do test cuts on scrap.


 Heres some work Ive done in the distant past.:

 The Starwars front was intentionally sanded a bit curved,
and light bondo covering cause I used crappy partical wood
which was too rough.   I was a bit of a novice at the time.

 Still, the angles had to be spot on for the thing to fit together...
which they were.

 The Top section is removable via screws.  The rest are screwed & glued in.

 The Super Hangon controller has an intentional Lip on the
edge.  I liked the look that way, instead of a flush cut mount.

[Akuma]

Again,   Unless you have some super accurate table saw,  your math
wont mean too much.

 How will you know your table saws angle is exactly 132 degrees for
example?   

 And, are you accounting for the blade width removal in your calculation?
Which side will it remove from... and or will it be centered removal?

 What if your saws guide is slightly off? 

 There are many issues.   mostly to do with material removal of the
blade thickness, and the less that adequate angle measurement on
most tablesaws.

 This is why its much easier and accurate to do test cuts on scrap.

You are right, I just encountered this. I ended up using alot of special glue which can also be used for minor modeling / bondo purposes and making the trapezoid more by guessing than calculating. It needs to dry now - I'll show you the finished piece when it's ready. Hope it will be satisfactory to me.

: )

[ScottS]

Again,   Unless you have some super accurate table saw,  your math
wont mean too much.

As a long-time woodworker and budding machinist, I would have to completely disagree with you! Accuracy of your table saw is almost completely irrelevant to this discussion. More important is the accuracy of your measuring tools. Anyone with $10-15 in their pocket can buy a Starrett angle meter that's accurate to a degree. If you've got another $5 or $10 you can probably score a machinist protractor that's twice as accurate. Either one is good enough to be able to setup a table saw or circular saw well enough to make cuts that are more than accurate enough.

Once you know the angles on your saw are correct, you just need to calculate how long your pieces need to be. There's no magic here. If you can measure the width of the top and bottom, you can use the formula I gave to calculate the length of the sides. Dealing with the saw kerf for these angles cuts is just like dealing with the saw kerf for any other cut! The kerf really only comes into play when you're laying out your cuts; if you want to get multiple pieces out of a single board, then the kerf becomes more important.

You're more than welcome to spend an afternoon in the garage hacking around on scrap wood and hoping that everything will work out when you scale things up to a full-size cabinet but for myself I'd rather do the math, make a single series of cuts, and have everything go together without needing a gallon of Bondo to mask the screw-ups. But that's just me...

[bfauska]

Again,   Unless you have some super accurate table saw,  your math
wont mean too much.

As a long-time woodworker and budding machinist, I would have to completely disagree with you! Accuracy of your table saw is almost completely irrelevant to this discussion. More important is the accuracy of your measuring tools. Anyone with $10-15 in their pocket can buy a Starrett angle meter that's accurate to a degree. If you've got another $5 or $10 you can probably score a machinist protractor that's twice as accurate. Either one is good enough to be able to setup a table saw or circular saw well enough to make cuts that are more than accurate enough.

Once you know the angles on your saw are correct, you just need to calculate how long your pieces need to be. There's no magic here. If you can measure the width of the top and bottom, you can use the formula I gave to calculate the length of the sides. Dealing with the saw kerf for these angles cuts is just like dealing with the saw kerf for any other cut! The kerf really only comes into play when you're laying out your cuts; if you want to get multiple pieces out of a single board, then the kerf becomes more important.

You're more than welcome to spend an afternoon in the garage hacking around on scrap wood and hoping that everything will work out when you scale things up to a full-size cabinet but for myself I'd rather do the math, make a single series of cuts, and have everything go together without needing a gallon of Bondo to mask the screw-ups. But that's just me...

+1

one change to above post.  As a long time woodworker/welder/designer/draftsman

[Akuma]

Okay, even if this seems to be slightly off topic now, I thought I might confront you guys with the result of the effort.

While I was aranging the wood with the original angle which was around 10 or 15 degree (bottom angles), I noticed that the angle would be too small (too small in design means, not in mathmatics means). After some thinking I decided to choose 30 and 60 degree angles which turned out to work greatly with the overall design since the screen and therefore the sides are angled at 30 degrees.

Final words: Sorry for all the comotion with an uncalculatable calculation which turned out to be not used after all : )

 

[bfauska]

Nice looking decision.  As a firm believer in figuring out with math, the one time I sway from that is when it is a matter of aesthetics, in those cases I believe in fidget until you find the look you're going for.  Again, most of my rant was academic, the info you gave was plenty for these purposes, and I'm used to figuring out problems to find that the solution was to change the whole approach anyway.

Looks like it'll be a cool cabinet when it's done.

[scotthh]

How did you put the curves on the front of the CP? And what is the circumference?

1st question is actually serious.

[Akuma]

Of course the cab is heavily work in progress : )

The curves were fairly easy. I used a divider set to the wood's strength and then used a jigsaw with a circle blade (you sure know that there are different jigsaw blades for making circular cuts aswell as for making straights). Afterwards, sanded smooth and voilá : )