The Right Triangle and the Pyramids Exploration

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1. Below are some Pythagorean triples. Compute the seqed - in this case b/a - in each case.

Triple (a, b, c) Corner Angle Seqed (b/a)
( 3, 4, 5) <math>53.1^o</math>
( 5, 12, 13) 67.4<math>^o</math>
( 7, 24, 25) <math>73.7^o</math>
( 8, 15, 17) <math>61.9^o</math>
( 9, 40, 41) <math>77.3^o</math>
(11, 60, 61) <math>79.6^o</math>
(12, 35, 37) <math>71.1^o</math>
(13, 84, 85) <math>81.2^o</math>
(16, 63, 65) <math>75.6^o</math>
(20, 21, 29) <math>46.4^o</math>
(28, 45, 53) 58.1<math>^o</math>
(33, 56, 65) <math>59.5^o</math>
(36, 77, 85) <math>64.9^o</math>
(39, 80, 89) <math>64.0^o</math>
(48, 55, 73) 48.9<math>^o</math>
(65, 72, 97) <math>47.9^o</math>


2. For the following pyramids do the following:
(i) Compute the Seqed if possible.
(ii)Could the pyramid have been constructed using a technique involving right triangles?

Pyramid Height base Seqed of the pyramid Slope Possible right triangle
Great Pyramid (Khufu) 146.6 230.3 51.8 degrees
Khafre 143.5 215.25 .75 53.1 degrees 3-4-5 Triangle
Menkaure 66.45 103.4 .79 51.3 degrees Maybe 3-4-5 Triangle?
Userkaf 49 73.3 53 degrees
Sahure 48 78.5 50.5 degrees
Queen Khentkawes 72 25 52 degrees
Djedkare 52.5 78.75 52 degrees
Djedkare's Queen 21 41 ca 45 degrees
Teti 52.5 78.5 53.1 degrees
Teti's Queen (Iput) 21 21 63 degrees


3. Note the interesting difference between the pyramids of the Kings and those of the Queens. It is quite common for the Queens to have a much smaller pyramid. The volume of a pyramid is given by <math>1/3 b^2 h</math>.
(i) Compute the volume the pyramids of Kings like Djedkare and Teti.
(ii) Compute the volumes of their queen's pyramids.
(iii) Given their volume, how much bigger are the king's monuments?