GeoGebra Classroom

AO-5677

Description

AO 5677 is a field plan from Umma (modern Tell Jokha), and was written during the Ur III period (2100-2000 BC). It currently rests in the Louvre Museum. More detail on this object can be found through the Cuneiform Digital Library Initiative.

Explination of the units and calculations

The lengths are given in nindan (1 nindan is approximately 6 meters), and the areas A1 through to A17 are given in iku (100 square nindan, or 3600 square meters, which is about half a football field). Each triangle is to be interpreted as a right triangle, and the area is calculated as half the base by the width. The area of a quadrilateral is calculated using the formula for the area of a trapezoid. The areas given above are calculated to two decimal places, whereas the areas recorded in the tablet itself are to the nearest quarter iku.

 ﻿Shape Actual Area (in square nindan)﻿ Recorded Area﻿ (in iku) ﻿A1 ﻿not recorded ﻿A2 6.25﻿ ﻿A3 ﻿ ﻿2.5 ﻿A4 ﻿ ﻿6.75 ﻿A5 ﻿24.75 ﻿A6 ﻿﻿﻿ ﻿5.25 - calculation error ﻿A7 ﻿﻿ ﻿4.5 ﻿A8 ﻿ ﻿1.5 ﻿A9 ﻿ ﻿1.25﻿ A10﻿ ﻿ 6 A11﻿ ﻿ 6 - calculation error A12﻿ ﻿ 3.75 A13 ﻿ 7.75 A14﻿ ﻿ 29.25 A15 ﻿ 3 A16 ﻿ 8 A17 ﻿ 2.5

Interpretation of the reverse

The reverse of the tablet lists the sub totals and grand total of the area of the field. ﻿ ﻿24 iku of small fields ﻿ ﻿80.25 iku of outside fields ﻿ ﻿201 iku of area within the temen (large central field). 305.25 total area The small fields are A6, A7, A8, A9, A10, A15 and A17. The field A15 is included as one of the small fields, as we shall later in the discussion of the temen, this inclusion was a mistake. The outside fields are A2, A3, A4, A5, A12, A13 and A14. The recorded area outside the fields is 80.25, which is different from the actual tally of the areas by 0.75 iku. The area within the temen is more interesting, and seems to have been calculated in the following manner. The top boundary was measured to 160, and the left boundary was measured to 130 which matches the right boundary. These top and left boundary lengths are clearly and prominently marked on the tablet. This is visualized as the beige quadrilateral in the image above, and has area of 208 iku. The area of the temen was then computed by adding three additional fields A11, A15 and A16 which lie outside this quadrilateral, and then subtracting the 'small fields' which lie inside it.