I think an important aspect of the design of Menkaure's pyramid is the visibility of the apex of Khufu's pyramid if you are standing on the top of Menkaure's pyramid:
![]()
Was the third pyramid planned so that the apex of G1 is visible over the lateral South-East edge of Khafre's pyramid?
Petrie's initial computation of G3's height is 65.1256 m. He also computed the height using the granite courses at 65.552 m. The height of G1 is 146.71 m, and the height of G2 is 143.56 m.
Maragioglio and Rinaldi computed the base levels of the three pyramids above sea level at 59.85 m for G1, 69.99 m for G2, and 71.42 m for G3.
The 65.1256 m G3 height value leads to an apex above sea level of 136.5456 m. The apex of G1 is computed to be 206.56 m.
Based on Petrie's survey the angle between East-G2 center base - G1 center base is 46.695 degrees.
Also, the G3-G2-G1 angle is 168.54 degrees.
Petrie computed the distance between G1-G2 to be 486.877 m, the G2-G3 distance is 453.976 m, and the G1-G3 distance is 936.158 m.
We want to compute the distance between the center base of G2 and the point that the SE diagonal of this pyramid horizontally intersects the G1 to G3 line. This length is y. The point is P. The angle between G1 to G2 to P is phi.
phi is equal to the angle East-G2-G1 which is 46.695 degrees plus the 45 degrees diagonal of a square so it is:
phi = 46.695 + 45 = 91.695 degrees
Using Petrie's data we can compute the G3-G1-G2 angle to be 5.5288 degrees. Since G1-G2 is 486.877 m we can compute trigonometrically the G1 - P distance to be 490.558 meters.
Subtracting this from the G1-G3 distance which is 936.158 m we get the distance between P and G3 to be 445.6 m.
From the apex altitude of G1 and G3 we compute the difference to be:
Da = (206.56 m)-(136.5456 m) = 70.0144 m
We compute the altitude increase from G3 to P:
i = (70.0144 m) x (445.6 m)/(936.158 m) = 33.326 m
This means that the altitude of the P position using the line of sight between G3 and G1 is:
ap = 33.326 m + 136.5456 m = 169.8716 m
With the previous information, we can compute trigonometrically the distance between G2 and P to be 47.284 m. We can compute the pyramid height that corresponds to the length:
h = 143.56 m - 143.56 m x (47.284 m)/(215.262 m/sqrt(2)) = 98.964 m
The altitude ae above sea level of this point of the lateral South-East edge of Khafre's pyramid is:
ae = h + 69.99 m + 98.964 m = 168.954 m
This means that the line of sight (G3-G1) is above the edge of Khafre's pyramid by a length of:
ap - ae = 169.8716 m - 168.954 m = 0.9176 meters = 1.75 royal cubits
The small size of this value seems to indicate that the size of Menkaure's pyramid was planned so as to lead to a marginal apex line of sight to the apex of Khufu's pyramid.
Today the capstones of both pyramids are missing and the present height of both is considerably smaller. It is not clear though if the line of sight exists due to the deterioration of the casing stones of Khafre's pyramid. It probably doesn't.
Was the third pyramid planned so that the apex of G1 is visible over the lateral South-East edge of Khafre's pyramid?
Petrie's initial computation of G3's height is 65.1256 m. He also computed the height using the granite courses at 65.552 m. The height of G1 is 146.71 m, and the height of G2 is 143.56 m.
Maragioglio and Rinaldi computed the base levels of the three pyramids above sea level at 59.85 m for G1, 69.99 m for G2, and 71.42 m for G3.
The 65.1256 m G3 height value leads to an apex above sea level of 136.5456 m. The apex of G1 is computed to be 206.56 m.
Based on Petrie's survey the angle between East-G2 center base - G1 center base is 46.695 degrees.
Also, the G3-G2-G1 angle is 168.54 degrees.
Petrie computed the distance between G1-G2 to be 486.877 m, the G2-G3 distance is 453.976 m, and the G1-G3 distance is 936.158 m.
We want to compute the distance between the center base of G2 and the point that the SE diagonal of this pyramid horizontally intersects the G1 to G3 line. This length is y. The point is P. The angle between G1 to G2 to P is phi.
phi is equal to the angle East-G2-G1 which is 46.695 degrees plus the 45 degrees diagonal of a square so it is:
phi = 46.695 + 45 = 91.695 degrees
Using Petrie's data we can compute the G3-G1-G2 angle to be 5.5288 degrees. Since G1-G2 is 486.877 m we can compute trigonometrically the G1 - P distance to be 490.558 meters.
Subtracting this from the G1-G3 distance which is 936.158 m we get the distance between P and G3 to be 445.6 m.
From the apex altitude of G1 and G3 we compute the difference to be:
Da = (206.56 m)-(136.5456 m) = 70.0144 m
We compute the altitude increase from G3 to P:
i = (70.0144 m) x (445.6 m)/(936.158 m) = 33.326 m
This means that the altitude of the P position using the line of sight between G3 and G1 is:
ap = 33.326 m + 136.5456 m = 169.8716 m
With the previous information, we can compute trigonometrically the distance between G2 and P to be 47.284 m. We can compute the pyramid height that corresponds to the length:
h = 143.56 m - 143.56 m x (47.284 m)/(215.262 m/sqrt(2)) = 98.964 m
The altitude ae above sea level of this point of the lateral South-East edge of Khafre's pyramid is:
ae = h + 69.99 m + 98.964 m = 168.954 m
This means that the line of sight (G3-G1) is above the edge of Khafre's pyramid by a length of:
ap - ae = 169.8716 m - 168.954 m = 0.9176 meters = 1.75 royal cubits
The small size of this value seems to indicate that the size of Menkaure's pyramid was planned so as to lead to a marginal apex line of sight to the apex of Khufu's pyramid.
Today the capstones of both pyramids are missing and the present height of both is considerably smaller. It is not clear though if the line of sight exists due to the deterioration of the casing stones of Khafre's pyramid. It probably doesn't.