The ancient Egyptians recognized that a triangle with sides of length 3, 4, and 5 units is a right triangle. They also recognized that a triangle with side lengths 6, 8 and 10 would also be right-angled because doubling the lengths of the sides would not change the shape of the triangle. This represented an important generalization that enabled them to create right angles using ropes of any length, thereby increasing the precision of their angles.
It is believed that the Egyptian rope stretchers made knots in long ropes to lay out the foundations of the pyramids to ensure that the base of the pyramid was square. That is, they would lay out the square base and check to ensure that the square of the diagonal length was the double the square of the side length.
The Pythagorean Theorem generalized the relationship between the length of the hypotenuse of a right-angled triangle and its sides as follows:
The square on the hypotenuse (longest side) of a right triangle has the same area as the sum of the squares drawn on the other two sides.
This theorem became the cornerstone for the construction of buildings that require square corners. Its generalization as the cosine law, later became a way of calculating lengths of the third side of any triangle given the length of two sides and the angle between them.
These mathematical laws are embedded in all engineering design programs. Furthermore, when a surveyor creates a map of all the lots in a neighborhood, these same theorems are used to check that the measured lengths and angles defining each lot are accurate. If the measured lengths and angles do not satisfy the cosine law, then we know there is an error in one of the measurements.
So, a mathematical relationship known to our earliest civilizations has become the foundation for the engineering of all our architectural foundations!
