Difference between revisions of "The Right Triangle and the Pyramids Exploration"

From EscherMath
Jump to navigationJump to search
(New page: 1. Below are some Pythagorean triples. Compute the seqed - in this case b/a - in each case. {| border="1" |- ! Triple (a, b, c) !! Corner Angle !! Seqed (b/a) |- | ( 3, 4, 5) || <math>...)
 
(category fix.)
 
(4 intermediate revisions by one other user not shown)
Line 1: Line 1:
 +
1. For the following pyramids compute the Seqed if possible. <br>
  
−
1. Below are some Pythagorean triples. Compute the seqed - in this case b/a - in each case.
 
 
{| border="1"
 
{| border="1"
 
|-
 
|-
−
! Triple (a, b, c) !! Corner Angle !! Seqed (b/a)
+
! Pyramid !! Height !! base !! Seqed of the pyramid !! Slope !! Possible right triangle
−
|-
 
−
| ( 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> ||
+
| Great Pyramid (Khufu) || 146.6 || 230.3 ||  || 51.8 degrees ||
 
|-
 
|-
−
| (13, 84, 85) || <math>81.2^o</math> ||
+
| Khafre || 143.5 || 215.25 || .75 || 53.1 degrees || 3-4-5 Triangle
 
|-
 
|-
−
| (16, 63, 65) || <math>75.6^o</math> ||
+
| Menkaure || 66.45 || 103.4 || .79 || 51.3 degrees || Maybe 3-4-5 Triangle?
 
|-
 
|-
−
| (20, 21, 29) || <math>46.4^o</math> ||
+
| Userkaf || 49 ||  73.3 ||  || 53 degrees ||  
 
|-
 
|-
−
| (28, 45, 53) || 58.1<math>^o</math> ||
+
| Sahure || 48 ||  78.5 ||  || 50.5 degrees ||  
 
|-
 
|-
−
| (33, 56, 65) || <math>59.5^o</math> ||
+
| Queen Khentkawes || 72 || 25 ||  || 52 degrees ||  
 
|-
 
|-
−
| (36, 77, 85) || <math>64.9^o</math> ||
+
| Djedkare || 52.5 ||  78.75 ||  || 52 degrees ||  
 
|-
 
|-
−
| (39, 80, 89) || <math>64.0^o</math> ||
+
| Djedkare's Queen || 21 ||  41 ||  || ca 45 degrees ||  
 
|-
 
|-
−
| (48, 55, 73) || 48.9<math>^o</math> ||
+
| Teti || 52.5 ||  78.5 ||  || 53.1 degrees ||  
 
|-
 
|-
−
| (65, 72, 97) || <math>47.9^o</math> ||
+
| Teti's Queen (Iput) || 21 ||  21 ||  || 63 degrees ||  
 
|-
 
|-
 
|}
 
|}
  
  
−
2. For the following pyramids determine if they could have been constructed using a technique involving right triangles.
 
  
−
{| border="1"
+
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>. <br>
−
! Pyramid !! Height !! base !! Seqed of the pyramid !! Slope !! Possible right triangle
+
(i) Compute the volume the pyramids of Kings like Djedkare and Teti. <br>
−
|-
+
(ii) Compute the volumes of their queen's pyramids. <br>
−
| Great Pyramid (Khufu) || 146.6 || 230.3 ||  || 51.8 ||
+
(iii) Given their volume, how much bigger are the king's monuments?
−
|-
+
 
−
| Khafre || 143.5 || 215.25 || .75 || 53.1 || 3-4-5 Triangle
+
[[Category:Egyptian Mathematics Explorations]]
−
|-
 
−
| Menkaure || 66.45 || 103.4 || .79 || 51.3 || Maybe 3-4-5 Triangle?
 
−
|-
 
−
|}
 

Latest revision as of 03:22, 10 February 2012

1. For the following pyramids compute the Seqed if possible.

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?

Retrieved from "http:///escher/index.php?title=The_Right_Triangle_and_the_Pyramids_Exploration&oldid=9378"