It is the proof that I am first modern to show slope of Great Pyramid with simple compass and rule method. Radians was one of the checks. Or can check by following problem. With a draw program you can get the angle very accuratly. But I was able to do that because I saw it in the new geometry.
Just able to copy directions from site, not down. Here:
Great Pyramid Slope By Construction
See Graphic. A check was done in radians. The links work, if you are rusty you can go to others to check.
Directions copies below from site that works.
These are directions for the proof of angle using simple rule and compass. However, because we are dealing with parts of degrees between 51 and 52 degrees a drawing or CAD program will be needed.
This explanation assumes the rest of the site is digested, which was the www.pavlita.com
which is now down.
So it must be understood that we are dealing in a 3-D geometry of equal sized balls, and because there are no lines or points a sphere by definition is impossible. However, to understand what is going on at times you must refer to concepts of plane geometry. We do here as it give a construction proof of the Great Pyramid Slope in a manner which is easy to understand as it is visual, not formulas. A ping pong ball model using 13 ping pong balls in the manner shown (there are two ways this works) would be helpful for the overall feeling, but not needed for this proof. At any rate you must first understand from 'book' (Letters Upon The Mast) four of the seven angles that the construction is sliced through, these are the four planes of the tetrahedron also. If the centers of the outside balls are connected, and lines are also drawn to the center of the center ball there are six pyramids and eight tetrahedrons exactly fitting together around a center. They must be understood as separate. A slope of each of the six pyramids shares a slope with four tetrahedrons. And this proof shows what happens when the sides of the six pyramids increase out (the slope is sixty degrees - is the pyramid's increase out that slope would decrease).
In other words all six pyramids are increasing in volume and the eight tetrahedrons are decreasing until they disappear. What then is the shape of EACH of the six pyramids? Will Show it is the exact Great Pyramid shape.
If you have trouble seeing the four rings, and how they would create the six sixty degree four sided slope pyramids and eight tetrahedrons (60 degrees also) around look perhaps also at the rings of the generator.
As you are creating this 'feel' the six pyramids expanding until the eight tetras disappear.
Though measurements here need to be exact, we can go directly to setting up a few exact ones to start. This makes the rest easy.
From experimentation I have discovered the right size to work this, too much accuracy is confusing, not enough does not give a picture.
Here we go:
Set the ruler to pixels. Make a hairline 6000 pixels high. This measurement and one sixth of it, 1000 pixels, are the critical measurements. I am assuming you are zooming and getting exact. Place this 6000 pixel line on a zero guideline. (be exact). Construct two angled guidelines, one of 60 and one of 120 degrees. Have them intersect with the top of the 6000 pixel line. Be exact. If the line is set at 'hairline' as it should be the width should still be almost a pixel. So have the lines cross in the middle. There is some zooming in and out quite a bit at the start. The 60 and 120 degree lines should now make an equilateral triangle. Use bezier tool and 'snap to guidelines' to make that shape. REMEMBER the lines are in the center of these guidelines. We will be measuring angles from the top of this triangle. You might make a crosshair very small at top. Something to show you that the very top of the shape is not the top, see where the centers of the lines meet. Now you need make a circle close to what the triangle would fit into if tip and center and two sides touching the circle. This does not need to be as exact. Get it close as you can. You can spend extra time getting it very exact but it will not buy you much. However, when placing over so that the top of triangle is in center and two bottom corners touch the circle, make sure the left side of the parts that touch the circle, (bottom left), is exact. Remembering again that the real center of the line is inside the approx. 3/4 of a pixel thick line understand that the sharp corner of the triangle will stick out a bit from the circle.
'Group' the work above or set in a layer so it does not come apart.
Now one more exact measure. This is 1/ 6th of the original 6000 pixel line. But we make it with a circle, showing various angles with this, so make a circle with an exact 2000 pixel diameter, and you may even want to put small crossshairs in the center. 'Group' this. Move this smaller circle, green in the diagram so that it sits direct on the center of where triangle crosses the greater circle. And when exact, group the entire thing. All the exact work is now done. So you have ballpark understanding, set, one at a time, four lines through the top of the pyramid. First do the 51.84444~ which I did in red. In yellow I did the parameters for understanding, I set one at 52 exact degrees and another at 52 exact degrees and did this in yellow. Also a darker yellow 52.73 degrees. Notice how the circle (smaller circle) cuts twice through the red, but about 51.73 at the greater circle.
1- As the six pyramids begin to move into, "eat up" the eight tetrahedrons, how for does each face of the pyramid move. It does not move to the center because there are five other moving as well. Each face need only move at 1/6th, the measurement of the radius of the lessor circle. The big triangle is the shape of the face of the six pyramids. It is also the shape of the base of the tetra exactly. So every face of each of the six pyramids moves from on side toward a tip. So the distance it moves is across, not one of the sides.
2- Notice that if it is considered that the base is the first move here, it is about 51.73.
3- However, we have the distance the face will move, 1000 pixels, but not the exact direction. As the pyramids are 'filling' up the tetras are decreasing. Therefore the angle that the bottom of the face moves is to be considered. The greater arc is the obvious limit. As they are all moving the other limit is half the 6000., 3000. Place lines of 3000 and 6000 so that they go from the corner of the pyramid to the arc. (I don't think the bottom blue line of 3000 comes through. However this angle is bisected which is the angle of second blue line. And you can see where that intersects the 51.84444~ given. Understand that the 51.844444, given by Taylor first is a formula based and theoretical. A visual of this concept has not yet been given except here.
Anyone who does this with a construction of 13 ping pong balls in hand will feel it as well as understand it.