Geometry wizzards - pyramid problem for ya

[Mayo]

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[Mayo]

OK I got this idea to make a little pyramid shape for the garden. I want to use 4 pieces of 2"x2" lumber which is in reality, only about 1.3125"

The base should be about 48" across and the height from the middle of the base to the point at the top should be about 42". The sticks are each about 48" long. (All measurements approximate - I'm not building a space shuttle)

At the top of the pyramid I need to cut the wood so that the four sticks assemble into a point. I tried 60 degrees cut off two sides of each stick as a guess and it was close but no cigar.

Any suggestions what angle to cut, or how to determine the angle for any given size pyramid?

After the four sticks are assembled, I had intended to put a masonite piece on each side at the top, 10 or 11" from the point to the middle of it's base, and cut to the same angle as the sides. Indicated by the red line.


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[gerald_d]

Hi Mayo, a 48x42 pyramid will need strips 54" long. The angle at top is actually a pair of flat faces at right angles to each other, if I understand your method correctly. The corner of the strip nearest the middle of the pyramid is 1.7" shorter and the two other corners to the left and right are 2.6" shorter. I would draw these two faces on the sticks and cut them freehand on a table or band saw. The ShopBot would be the last tool I would consider for this - unless you make a jig to hold the stick over the edge of the table, in the orientation you show in your sketch.

The masonite will not have flat surfaces to nail on to.

(If you want to use a single cut on the stick face, and let the masonite hide the gaps, you can slant it back to 5.2" from the tip)

[kaaboom_99]

Hi Mayo, Love a puzzle once in a while. First, look at your drawing as 2 triangles, back to back, with an imaginary line drawn from top to bottom. Now you have two triangle with a right angle in each, each with a base of 24" and a side that is 42". Using the formulae for SAS, I get an angle at the top (for 1 triangle) of 29.74488129694222 degrees which when double gives you Approx 59.5 degrees. The outside pieces are around 48.373546489791295 inches long and the angle at the base of this outside piece and ground will be about 60.255118703057775 degrees. I hope this helps. You can run the numbers if you go to this website http://www.1728.com/trig4.htm.

[kaaboom_99]

Sorry Mayo, I said I love puzzles, didn't mean I was any good. That aside, I double checked my numbers, the above post is bogus. Go to that website and use the S-S-S method (in your case 48 x 48 x 42) in which case the angle at the top is 51.8889595447400 degrees - not quite 60 just as you thought. Sorry about the bogus info. After 12 hours of work, I should probably keep my mouth shut.
Hope these numbers work for you.
Perry

[gerald_d]

Two facets at 90 degrees to each other:


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[gerald_d]

"Cheat" cut with one long facet:


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[Mayo]

Thanks for the help and suggestions so far!

I cut the 60 degree angles on a table saw - I agree this particular cut is not for the shopbot. I did, however, cut my masonite pyramid top pieces on the shopbot.

Just to further clarify what I had wanted, each side of the pyramid will have the sticks joined at the top so that they create a flush surface or plane. One stick does not lay on top of the other. There is no gap. The masonite will lay flat on each side of the pyramid. I'll make two opposing sides of these masonite pieces slightly larger so that they overlap the thickness of the masonite on the other sides, like a butt joint.

As the four sticks lay together at the top, only their inside surfaces are cut on whatever angle that will allow the pyramid to be the size indicated.

My first attempt used sticks that are only 48" long and they assemble so that they are about 48" apart at the bottom. The top point is about 42.5" up from the middle of the base (using the incorrect 60 degree cuts). 54" sticks are too long.


4717

[beacon14]

Using the formula for a right triangle with base 24" and hypotenus 48" (one half of each side of your pyramid), the height of each side would be 42 9/16" if the side were vertical, and the angle at the top would be 60 degrees (since you have an equilateral triangle, 48" on each side).
However, since each side leans into the center at the top (by 35.3 degrees), according to a simple CAD sketch I did, your resulting height of the pyramid would be 33 15/16".
My 3d angle skills are too rusty to calculate the angles required (sorry, I know this is what you wanted to know in the first place).

Gerald has a good point, that as you lean each side into the center, the legs "twist", so that there are no flat planes for the masonite to bed against, since each leg serves as an upright for two sides.

Another way to build it might be to make four 3-sided triangles, complete with masonite, then lean them together at the top. If you rip each leg at a bevel along it's entire length, so that as two sides come together in a corner the legs mate up, you would have your 4 flat sides for the masonite and fairly strong construction. Of course, this requires twice as many 2x2s, but the bevel angle shouldn't be too hard to figure out with trial and error.

[gerald_d]

Mayo, your figures seem to be based on a single side face of the pyramid. It looks like you are taking 2 sticks, laying them flat on a bench, and nailing on the masonite. Once you have two of these, then you lean them together to form the pyramid? (Which will then only be 35" high?)

If your masonite is a good job, why bother to run the sticks all the way into the apex? Suggest that you cut the sticks off square and let the masonite (plus some expoxy) do the rest.

To see the true length of a stick, you must look at your pyramid from direction A:


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[tberkner]

A common woodworking problem

http://www.josephfusco.org/Articles/Dihedral/Dihedral1.html this is a good explanation

http://www.scarletta.net/WoodJoy/compound_miter.html this is a program that will calculate the angles for you