Ancient Egypt, Old Kingdom
2585 BC–2560 BC
The design and build of each Egyptian pyramid was overseen by a royal master builder.
Charged with creating a symmetrical masterpiece, he had to find a way to create equal sides and angles without complex instruments.
It is thought that master builders might have relied on the inherent qualities of a certain type of triangle.
Called a 3-4-5 triangle, its sides always conform to this ratio.
And, crucially, it is always a right triangle, with one corner measuring a precise 90°.
Because the Egyptians began the construction of every pyramid by creating a square base, 3-4-5 triangles were very useful.
They could ensure that a true square was formed, with 90° corners, rather than a rhombus.
The base of the Great Pyramid at Khufu, for instance, was created from triangles measuring 21, 28 and 35 cubits - simply 3-4-5 triangles multiplied by seven!
Angle of Incline
It is not only the right angle in a 3-4-5 triangle, which is fixed.
Both of the other angles are always the same.
This allowed master builders to ensure similar angles of incline throughout their pyramids.
Pyramids of Giza
Present day Egypt
When a perpendicular bisector cuts from the apex to the foundation of the Pyramid of Khufu - it creates two 3-4-5 triangles.
Because the angle of incline for a 3-4-5 triangle is always 53.13°, the incline is therefore the same on both sides.
Most Egyptian pyramids have a 53° incline.
The Bent Pyramid 2600 BC
But not the Bent Pyramid.
Its angle of incline had to be lowered to just 43° in the upper section.
In order to get the 43° angle they may have used a triangle with the ratio of 20, 21 and 29.
It is thought that the angle of incline was reduced because, part-way through construction, the pyramid seemed unstable.
So the royal master builder had to revisit his design.