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Re: Ariadne's thread
Posted:
Feb 28, 2012 3:07 AM
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RMP 50 on the advanced level
Ahmes calculates the area of a circle whose diameter
measures 9 khet = 900 royal cubits. By using his well
known formula he obtains 8 by 8 khet = 64 square khet
= 64 aroures or setat = 640,000 square cubits.
Advanced learners may try to solve a more demanding task:
transforming the circle into a regular octagon of the
same area using the following extended number pattern:
1 1 2 2
2 3 4 6
5 7 10 14
12 17 24 34
29 41 58 82
70 99 140 198
169 239 338 478
The squared side of a regular octagon and the area of the
same octagon maintain a relation that can be approximated
by means of the above numbers:
side x side 12 17 29 41 70 99 ... square cubits
area octagon 58 82 140 198 338 478 ... square cubits
The pi value may be chosen from the following sequence:
3/1 (plus 22/) 25/8 47/15 69/22 91/29 113/36 135/43
157/50 179/57 201/64 223/71 245/78 267/85 289/92
311/99 333/106 355/113 377/120 399/127
Two values contain the number 99: '99 of 478 and '99 of 311.
Now the area of a regular octagon and the one of a circle
may be defined like this:
side x side x '99 x 478, radius x radius x '99 x 311
The octagon and the circle have the same area, therefore:
side x side x '99 x 478 = radius x radius x '99 x 311
The diameter of the circle measures 9 khet or 900 royal
cubits while the radius measures 450 royal cubits.
Now we obtain:
side x side = 450 cubits x 450 cubits x 311 x '478
side x side = practically 131,752 square cubits
By consulting a table of square numbers you will find
362 x 362 = 131,044 --- 708 less than 131,752
363 x 363 = 131,769 --- only 17 more than 131,752
364 x 364 = 132,496 --- 744 more than 131,752
The number 363 is a good solution to our problem. Hence
a circle of the diameter 9 khet and a regular octagon
of the side length 363 royal cubits have practically
the same area.
grid 770+1089+770 by 770+1089+770, unit '3 royal cubit
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