Frame Triangles and the Power of Pythagoras

​​I rushed my Klann Ver 2 build and didn't build the outer frames in an optimal way:

The only thing keeping this frame's corners at right angles are the two 3x5 L-shaped LEGO parts.  If this frame were put under a lot of force, like would happen at a larger scale, the corners would be subjected to torque that could easily cause the rectangle to collapse. To avoid this diagonals need to be added, which will convert the rectangles into triangles and lock the corner's angles.
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The challenge with walkers in LEGO is we often need diagonals for rectangles that define the linkage's parameters.  In other words, these diagonals often need to be the hypotenuses of right triangles.   As you can see below, my Klann's upper support rods are 7 holes above the motor's axle, and the lower support rods are 2 holes below the motor's axle.  Neither of these lengths work with a 3-4-5 or 6-8-10 right triangle with LEGO parts for hypotenuses. What can we do using the integer lengths of LEGO's beams?  

NOTE:  When determining the length of LEGO beams the first hole is always counted as 0!  If you don't measure LEGO beam lengths in this way you won't be able to use the Pythagorean theorem to calculate which beams to use as hypotenuses.
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Fortunately, with LEGO we don't have to be at precise integer numbers, and we can use hypotenuses that are close enough. 
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1.  Hypotenuses for rectangles of height 7. 
Plugging in a 90 degree angle with a side length of 7, plus a few other whole number sides into the Pythagorean theorem yields this near integer number triangle:

​I used this triangle for Klann's inner frame where two 9 hole beams create hypotenuses of length 8:

I also used this triangle for TrotBot's frame:

​2. Hypotenuses for rectangles of height 9. 
We can also connect the top and bottom beams of Klann's frame with a hypotenuse:

Plugging a 90 degree angle with a side length of 9, plus a few other whole number sides into the Pythagorean theorem yields another near integer number triangle:

So, 13 hole LEGO beams can be used to lock Klann's outer frames into triangles, like this:

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Here are a few other useful triangles for LEGO frames:

​​Also, the below 5-3 bent beam can be used as a hypotenuse.

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​The length of the part is 7.212:  

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​Which allows us to create this right triangle: 

You can create more triangles by lengthening a side of the above parts by attaching a LEGO beam to it.

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​As shown in the image to the right, Strider's frame height is not a whole number, but is 9.165.   Below shows some LEGO beam lengths for creating frames of a few different heights.

LEGO beam lengths for creating frames of varying heights