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

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

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− | 2. For the following pyramids | + | 2. For the following pyramids do the following: <br> |

+ | (i) Compute the Seqed if possible. <br> | ||

+ | (ii)Could the pyramid have been constructed using a technique involving right triangles? | ||

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! Pyramid !! Height !! base !! Seqed of the pyramid !! Slope !! Possible right triangle | ! Pyramid !! Height !! base !! Seqed of the pyramid !! Slope !! Possible right triangle | ||

|- | |- | ||

− | | Great Pyramid (Khufu) || 146.6 || 230.3 || || 51.8 || | + | | 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 || || 62 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 with a much steeper slope. The volume of a pyramid is given by <math>1/3 b^2 h</math>. <br> | ||

+ | (i) Compute the volume the pyramids of Kings like Djedkare and Teti. <br> | ||

+ | (ii) Compute the volumes of their queen's pyramids. <br> | ||

+ | (iii) Given their volume, how much bigger are the king's monuments? |

## Revision as of 16:03, 17 November 2008

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 | 62 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 with a much steeper slope. 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?