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**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.

**3-4-5 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.