# Problems

### Area of triangles and rectangles.

1. a. What is the area formula for the triangle?

b. One of the problmes on the Rhind papyrus asks: *What is the area of a triangle of height 10 khet and a base of 4 khet?* Compute the area.

2. a. What is the area formula for a rectangle?

b. One of the papyri has the problem *Find the area of a rectangular plot of land measuring 10 khet by 2.*. FInd the area.

3. The Egyptians had a slightly different way of writing up solutions.

[line 1] Example of Calculating a rectangle

[line 2] If someone says to you: A rectangle is 12 setjat [in area] [has a breadth 1/2 1/4 [i.e. 3/4] of its length.[Calculate it]

[line 3] Calculate 1/2 1/4 to get 1. The result is 1 1/3

[line 4] Take this 12 setjat 1 1/3 times. The result is 16.

[line 5] Calculate its square root. The result is 4 for its length [and] 1/2 1/4 of it is 3 for the breadth

a. Write down the question as we would ask it.

b. How would we write up the answer if we needed to hand in a detailed solution?

### Circles

4. In one of the papyri the computation of the area of a circle leads to the fact that *... the number <math>4(\frac{8}{9})^2 = 3 \frac{13}{81}</math> plays the role of π.* How accurate is this approximation of π?

### 3-Dimensional shapes

5. We compute the volume of a cylinder of radius R and height H to be: Vol = π <math>R^2*H</math>.

Compute the volume of a cylindrical granary of diameter 9 and height 10. (Rhind papyrus nr 41)

6. We compute the volume of a rectangular perallelopiped (a box) of length L, width W and height H to be: Vol = <math> L*W*H</math>.

Compute the volume of a rectangular granary of length 10, breadth 10 and height 10. (Rhind papyrus nr 44)

### General

7.

Pyramid | Height | base |
---|---|---|

Great Pyramid (Khufu) | 146.6 | 230.3 |

Khafre | 143.5 | 215.25 |

Menkaure | 66.45 | 103.4 |

Userkaf | 49 | 73.3 |

Sahure | 48 | 78.5 |

Queen Khentkawes | 72 | 25 |

Djedkare | 52.5 | 78.75 |

Djedkare's Queen | 21 | 41 |

Teti | 52.5 | 78.5 |

Teti's Queen (Iput) | 21 | 21 |

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?