The Right Triangle and the Pyramids Exploration

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?