CALCULATING THE HEIGHT OF CHEOPS PYRAMID
Thales of Miletus
• Around the 620 BC • Miletus was an ancient Greek city on the western coast of Anatolia in what is now Turkey • Philosopher, astronomer, geographer, mathematician… • Great observer and traveller.
• In one of his travels to Egypt he aimed to measure the height of the Great Pyramid of Cheops. • There are many theories about how he accomplished his goal but there is agreement that he used the concept of similarity of triangles.
Triangles are similar if they have the same shape, but not necessarily the same size. Have all the same angles. Its correspondent sides are proportional.
PQ QR PR P 'Q ' Q ' R ' P ' R '
17 15 12 2 8,5 7,5 6
Two right angled triangles are similar when one of their angles other than the rigth ones (90º) are equal in measure. Isoceles right angled triangles
The sun is so far away that we might consider its rays parallel and so, they hit the tops of all the objects at the same slant.
h = Thales’ height H = Height of the pyramid
H h h Pyramid S (shade)
What was really new?
• • Do you see what Thales did? He used an abstract right triangle! He pictured the height of the Great Pyramid as an imaginary post from its top straight down to its base. Such an imaginary post would cast an imaginary shadow, all the way from where it stood at the center of the pyramid to the tip of the pyramid's real shadow: so the length of this imaginary shadow would be the length of the base plus the actual projecting shadow!