20:
1754:
1706:
1743:
1774:
161:
1784:
77:
1764:
157:. These cubits are 52.5 cm (20.7 in) long and are divided into palms and hands: each palm is divided into four fingers from left to right and the fingers are further subdivided into ro from right to left. The rules are also divided into hands so that for example one foot is given as three hands and fifteen fingers and also as four palms and sixteen fingers.
648:
Problem 50 of the RMP finds the area of a round field of diameter 9 khet. This is solved by using the approximation that circular field of diameter 9 has the same area as a square of side 8. Problem 52 finds the area of a trapezium with (apparently) equally slanting sides. The lengths of the parallel
468:
in London. The problem also demonstrates that the
Egyptians were familiar with square roots. They even had a special hieroglyph for finding a square root. It looks like a corner and appears in the fifth line of the problem. Scholars suspect that they had tables giving the square roots of some often
72:
The ancient
Egyptians wrote out their problems in multiple parts. They gave the title and the data for the given problem, in some of the texts they would show how to solve the problem, and as the last step they verified that the problem was correct. The scribes did not use any variables and the
1058:
The Nile occupied an important position in
Egyptian culture; it influenced the development of mathematics, geography, and the calendar; Egyptian geometry advanced due to the practice of land measurement "because the overflow of the Nile caused the boundary of each person's land to
644:
That this octagonal figure, whose area is easily calculated, so accurately approximates the area of the circle is just plain good luck. Obtaining a better approximation to the area using finer divisions of a square and a similar argument is not simple.
676:
Several problems compute the volume of cylindrical granaries (41, 42, and 43 of the RMP), while problem 60 RMP seems to concern a pillar or a cone instead of a pyramid. It is rather small and steep, with a seked (slope) of four palms (per cubit).
463:
Problem 49 from the RMP finds the area of a rectangular plot of land
Problem 6 of MMP finds the lengths of the sides of a rectangular area given the ratio of the lengths of the sides. This problem seems to be identical to one of the
129:
rods. Examples have been found in the tombs of officials, noting lengths up to remen. Royal cubits were used for land measures such as roads and fields. Fourteen rods, including one double-cubit rod, were described and compared by
176:
shows surveyors measuring a plot of land using rope with knots tied at regular intervals. Similar scenes can be found in the tombs of
Amenhotep-Sesi, Khaemhat and Djeserkareseneb. The balls of rope are also shown in
857:
359:
990:
638:
554:
1262:
Clagett, Marshall
Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999
769:
665:
446:
249:
387:
905:
587:
289:
684:
computes the volume of a granary with a circular base. A similar problem and procedure can be found in the Rhind papyrus (problem 43). Several problems in the
1006:. Such a formula would be needed for building pyramids. In the next problem (Problem 57), the height of a pyramid is calculated from the base length and the
483:
Problem 48 of the RMP compares the area of a circle (approximated by an octagon) and its circumscribing square. This problem's result is used in problem 50.
64:(RMP). The examples demonstrate that the ancient Egyptians knew how to compute areas of several geometric shapes and the volumes of cylinders and pyramids.
107:
diagram shows how to construct a circular vault using body measures along an arc. If the area of the Square is 434 units. The area of the circle is 433.7.
1002:
Problem 56 of the RMP indicates an understanding of the idea of geometric similarity. This problem discusses the ratio run/rise, also known as the
1400:
487:
Trisect each side. Remove the corner triangles. The resulting octagonal figure approximates the circle. The area of the octagonal figure is:
476:
A translation of the problem and its solution as it appears on the fragment is given on the website maintained by
University College London.
784:
1426:
1498:
1010:(Egyptian for slope), while problem 58 gives the length of the base and the height and uses these measurements to compute the seked.
122:. A curve is divided into five sections and the height of the curve is given in cubits, palms, and digits in each of the sections.
1727:
1555:
305:
1355:
1267:
1044:
474:
An area of 40 "mH" by 3 "mH" shall be divided in 10 areas, each of which shall have a width that is 1/2 1/4 of their length.
469:
used numbers. No such tables have been found however. Problem 18 of the MMP computes the area of a length of garment-cloth.
1673:
1034:
56:
We only have a limited number of problems from ancient Egypt that concern geometry. Geometric problems appear in both the
1787:
1545:
1540:
1421:
168:
Surveying and itinerant measurement were undertaken using rods, poles, and knotted cords of rope. A scene in the tomb of
1393:
921:
594:
1510:
695:
Problem 14 of the Moscow
Mathematical Papyrus computes the volume of a truncated pyramid, also known as a frustum.
492:
103:, when the height of the Nile was recorded as 6 cubits and 1 palm (about 3.217 m or 10 ft 6.7 in). A
1663:
1431:
1695:
1641:
1493:
725:
1285:
R.C. Archibald
Mathematics before the Greeks Science, New Series, Vol.71, No. 1831, (Jan. 31, 1930), pp.109-121
1767:
1668:
1232:
1203:
1178:
1153:
1113:
85:
1808:
1454:
1449:
1386:
1813:
1601:
1476:
1444:
1685:
1648:
1596:
1515:
1015:
If you construct a pyramid with base side 12 and with a seked of 5 palms 1 finger; what is its altitude?
1013:
In
Problem 59 part 1 computes the seked, while the second part may be a computation to check the answer:
685:
669:
57:
412:
215:
1535:
689:
61:
24:
1658:
1626:
1611:
1606:
1505:
1459:
681:
465:
366:
73:
problems were written in prose form. The solutions were written out in steps, outlining the process.
1777:
1653:
1574:
1525:
1757:
1690:
1530:
872:
104:
559:
1722:
1221:
1747:
1705:
1562:
1471:
1464:
1365:
178:
265:
8:
1567:
1488:
1621:
1584:
1483:
1317:
1300:
1579:
1520:
1351:
1263:
1228:
1199:
1174:
1149:
1109:
1040:
456:= height. Calculations of the area of a triangle appear in both the RMP and the MMP.
672:. The problem includes a diagram indicating the dimensions of the truncated pyramid.
49:
to preserve the layout and ownership of farmland, which was flooded annually by the
1636:
1550:
1333:
1311:
1294:
111:
1369:
1337:
1091:, Architecture and Mathematics in Ancient Egypt, Cambridge University Press, 2007
139:
131:
1616:
1802:
1409:
1088:
173:
154:
89:
38:
19:
1678:
115:
143:
1717:
1378:
93:
50:
46:
852:{\displaystyle V={\frac {32}{27}}d^{2}\ h={\frac {128}{27}}r^{2}\ h}
160:
1589:
182:
42:
34:
692:(numbers 44, 45, 46) compute the volume of a rectangular granary.
76:
1631:
664:
260:
Problem 49 in RMP and problems 6 in MMP and Lahun LV.4. problem 1
135:
119:
97:
1003:
649:
sides and the distance between them being the given numbers.
169:
126:
100:
1350:, Vol. 232. Philadelphia: American Philosophical Society.
656:
Problem 10 of the MMP computes the area of a hemisphere.
409:
The ancient Egyptians knew that the area of a triangle is
362:
d= diameter. This uses the value 256/81 = 3.16049... for
150:
354:{\displaystyle A={\frac {1}{4}}({\frac {256}{81}})d^{2}}
1122:
924:
875:
787:
728:
597:
562:
495:
415:
369:
308:
268:
218:
1033:
Erlikh, Ḥagai; Erlikh, Hạggai; Gershoni, I. (2000).
1032:
1220:
984:
899:
851:
763:
632:
581:
548:
440:
381:
353:
300:Problems 51 in RMP and problems 4, 7 and 17 in MMP
283:
243:
1281:
1279:
1277:
1275:
985:{\displaystyle V={\frac {1}{3}}(a^{2}+ab+b^{2})h}
472:The Lahun Papyrus Problem 1 in LV.4 is given as:
210:Problem 51 in RMP and problems 4, 7 and 17 in MMP
1800:
633:{\displaystyle 4({\frac {8}{9}})^{2}=3.16049...}
84:Egyptian units of length are attested from the
1272:
1198:. Griffith Institute Asmolean Museum, Oxford.
1106:Ancient Egyptian Construction and Architecture
556:Next we approximate 63 to be 64 and note that
549:{\displaystyle 9^{2}-4{\frac {1}{2}}(3)(3)=63}
1394:
1371:Die Alt-Aegyptische Elle und Ihre Eintheilung
1039:. Lynne Rienner Publishers. pp. 80–81.
680:A problem appearing in section IV.3 of the
125:At some point, lengths were standardized by
88:. Although it dates to the 5th dynasty, the
764:{\displaystyle V={\frac {256}{81}}r^{2}\ h}
1401:
1387:
1258:
1256:
1254:
1252:
1250:
1248:
1246:
1244:
1339:Ancient Egyptian Science: A Source Book,
1103:
1408:
1218:
1193:
1168:
1143:
663:
159:
75:
18:
1364:
1332:
1241:
1223:Mathematics in the Time of the Pharaohs
1196:A Concise Dictionary of Middle Egyptian
1128:
1073:
96:during the reign of the Early Dynastic
1801:
149:Another was found in the tomb of Kha (
1382:
1763:
1139:
1137:
1099:
1097:
1084:
1082:
1069:
1067:
1036:The Nile: Histories, Cultures, Myths
1783:
1315:Digitalegypt website: Lahun Papyrus
1298:Digitalegypt website: Lahun Papyrus
13:
1212:
908:w = width, l = length, h = height
441:{\displaystyle A={\frac {1}{2}}bh}
244:{\displaystyle A={\frac {1}{2}}bh}
134:. Two examples are known from the
14:
1825:
1511:Ancient Egyptian race controversy
1187:
1134:
1094:
1079:
1064:
640:plays the role of π = 3.14159....
185:, Amenemhet-Surer, and Penanhor.
1782:
1772:
1762:
1753:
1752:
1741:
1704:
712:Formula (using modern notation)
202:Formula (using modern notation)
164:Cubit rod from the Turin Museum.
37:as it was developed and used in
1773:
1326:
1305:
1288:
382:{\displaystyle \pi =3.14159...}
112:ostracon depicting this diagram
1173:. Oxford: Griffith Institute.
1162:
1026:
976:
941:
615:
601:
537:
531:
528:
522:
338:
325:
1:
1374:(in German). Berlin: DĂĽmmler.
1019:
668:Image of Problem 14 from the
181:statues of officials such as
45:was a necessary outgrowth of
16:Geometry emanating from Egypt
1343:Ancient Egyptian Mathematics
1171:Egyptian Grammar 3rd Edition
7:
1696:Egypt–Mesopotamia relations
1516:Population history of Egypt
913:Truncated pyramid (frustum)
697:
686:Moscow Mathematical Papyrus
670:Moscow Mathematical Papyrus
187:
58:Moscow Mathematical Papyrus
10:
1830:
1219:Gillings, Richard (1972).
1194:Faulkner, Raymond (1991).
1144:Loprieno, Antonio (1996).
1104:Englebach, Clarke (1990).
690:Rhind Mathematical Papyrus
659:
92:recorded the level of the
62:Rhind Mathematical Papyrus
25:Rhind Mathematical Papyrus
1736:
1713:
1702:
1440:
1417:
900:{\displaystyle V=w\ l\ h}
771:measured in cubic-cubits
682:Lahun Mathematical Papyri
466:Lahun Mathematical Papyri
1748:Ancient Egypt portal
1169:Gardiner, Allen (1994).
997:
688:(problem 14) and in the
582:{\displaystyle 64=8^{2}}
67:
986:
901:
853:
765:
673:
634:
583:
550:
442:
383:
355:
292:b = base, h = height
285:
252:b = base, h = height
245:
165:
81:
27:
1422:Glossary of artifacts
1366:Lepsius, Karl Richard
987:
902:
864:Rectangular granaries
854:
776:Cylindrical granaries
766:
717:Cylindrical granaries
667:
635:
584:
551:
443:
384:
356:
286:
246:
163:
86:Early Dynastic Period
79:
22:
1809:Egyptian mathematics
922:
873:
867:RMP 44-46 and MMP 14
859:(measured in khar).
785:
726:
595:
560:
493:
413:
367:
306:
284:{\displaystyle A=bh}
266:
216:
1814:History of geometry
1568:Cursive hieroglyphs
1108:. New York: Dover.
702:
192:
142:, the treasurer of
114:was found near the
1541:Funerary practices
1348:Memoirs of the APS
982:
897:
849:
779:RMP 42, Lahun IV.3
761:
698:
674:
630:
579:
546:
438:
379:
351:
281:
241:
188:
166:
82:
28:
1796:
1795:
1551:Great Royal Wives
1521:Prehistoric Egypt
1357:978-0-87169-232-0
1334:Clagett, Marshall
1268:978-0-87169-232-0
1148:. New York: CUP.
1131:, pp. 57 ff.
1046:978-1-55587-672-2
995:
994:
939:
893:
887:
845:
831:
816:
802:
757:
743:
612:
520:
430:
403:
402:
397:Problem 10 in MMP
336:
323:
233:
60:(MMP) and in the
31:Egyptian geometry
1821:
1786:
1785:
1776:
1775:
1766:
1765:
1756:
1755:
1746:
1745:
1744:
1708:
1403:
1396:
1389:
1380:
1379:
1375:
1361:
1320:
1316:
1312:Annette Imhausen
1309:
1303:
1299:
1295:Annette Imhausen
1292:
1286:
1283:
1270:
1260:
1239:
1238:
1226:
1216:
1210:
1209:
1191:
1185:
1184:
1166:
1160:
1159:
1146:Ancient Egyptian
1141:
1132:
1126:
1120:
1119:
1101:
1092:
1086:
1077:
1071:
1062:
1061:
1055:
1053:
1030:
991:
989:
988:
983:
975:
974:
953:
952:
940:
932:
906:
904:
903:
898:
891:
885:
858:
856:
855:
850:
843:
842:
841:
832:
824:
814:
813:
812:
803:
795:
770:
768:
767:
762:
755:
754:
753:
744:
736:
703:
639:
637:
636:
631:
623:
622:
613:
605:
591:Thus the number
588:
586:
585:
580:
578:
577:
555:
553:
552:
547:
521:
513:
505:
504:
447:
445:
444:
439:
431:
423:
388:
386:
385:
380:
360:
358:
357:
352:
350:
349:
337:
329:
324:
316:
290:
288:
287:
282:
250:
248:
247:
242:
234:
226:
193:
1829:
1828:
1824:
1823:
1822:
1820:
1819:
1818:
1799:
1798:
1797:
1792:
1742:
1740:
1732:
1709:
1700:
1436:
1413:
1407:
1358:
1329:
1324:
1323:
1314:
1310:
1306:
1297:
1293:
1289:
1284:
1273:
1261:
1242:
1235:
1217:
1213:
1206:
1192:
1188:
1181:
1167:
1163:
1156:
1142:
1135:
1127:
1123:
1116:
1102:
1095:
1087:
1080:
1072:
1065:
1051:
1049:
1047:
1031:
1027:
1022:
1000:
970:
966:
948:
944:
931:
923:
920:
919:
907:
874:
871:
870:
837:
833:
823:
808:
804:
794:
786:
783:
782:
749:
745:
735:
727:
724:
723:
662:
655:
642:
618:
614:
604:
596:
593:
592:
573:
569:
561:
558:
557:
512:
500:
496:
494:
491:
490:
482:
462:
422:
414:
411:
410:
408:
368:
365:
364:
361:
345:
341:
328:
315:
307:
304:
303:
291:
267:
264:
263:
251:
225:
217:
214:
213:
80:Egyptian circle
70:
17:
12:
11:
5:
1827:
1817:
1816:
1811:
1794:
1793:
1791:
1790:
1780:
1770:
1760:
1750:
1737:
1734:
1733:
1731:
1730:
1725:
1720:
1714:
1711:
1710:
1703:
1701:
1699:
1698:
1693:
1688:
1683:
1682:
1681:
1676:
1666:
1661:
1656:
1651:
1646:
1645:
1644:
1639:
1629:
1624:
1619:
1614:
1609:
1604:
1599:
1594:
1593:
1592:
1587:
1577:
1572:
1571:
1570:
1560:
1559:
1558:
1548:
1543:
1538:
1533:
1528:
1523:
1518:
1513:
1508:
1503:
1502:
1501:
1491:
1486:
1481:
1480:
1479:
1469:
1468:
1467:
1462:
1457:
1447:
1441:
1438:
1437:
1435:
1434:
1429:
1424:
1418:
1415:
1414:
1406:
1405:
1398:
1391:
1383:
1377:
1376:
1362:
1356:
1328:
1325:
1322:
1321:
1304:
1287:
1271:
1240:
1233:
1211:
1204:
1186:
1179:
1161:
1154:
1133:
1129:Lepsius (1865)
1121:
1114:
1093:
1078:
1074:Clagett (1999)
1063:
1045:
1024:
1023:
1021:
1018:
999:
996:
993:
992:
981:
978:
973:
969:
965:
962:
959:
956:
951:
947:
943:
938:
935:
930:
927:
917:
914:
910:
909:
896:
890:
884:
881:
878:
868:
865:
861:
860:
848:
840:
836:
830:
827:
822:
819:
811:
807:
801:
798:
793:
790:
780:
777:
773:
772:
760:
752:
748:
742:
739:
734:
731:
721:
718:
714:
713:
710:
707:
661:
658:
629:
626:
621:
617:
611:
608:
603:
600:
576:
572:
568:
565:
545:
542:
539:
536:
533:
530:
527:
524:
519:
516:
511:
508:
503:
499:
485:
437:
434:
429:
426:
421:
418:
401:
400:
398:
395:
391:
390:
378:
375:
372:
348:
344:
340:
335:
332:
327:
322:
319:
314:
311:
301:
298:
294:
293:
280:
277:
274:
271:
261:
258:
254:
253:
240:
237:
232:
229:
224:
221:
211:
208:
204:
203:
200:
197:
69:
66:
15:
9:
6:
4:
3:
2:
1826:
1815:
1812:
1810:
1807:
1806:
1804:
1789:
1781:
1779:
1771:
1769:
1761:
1759:
1751:
1749:
1739:
1738:
1735:
1729:
1726:
1724:
1723:Egyptologists
1721:
1719:
1716:
1715:
1712:
1707:
1697:
1694:
1692:
1689:
1687:
1684:
1680:
1677:
1675:
1672:
1671:
1670:
1667:
1665:
1662:
1660:
1657:
1655:
1652:
1650:
1647:
1643:
1640:
1638:
1635:
1634:
1633:
1630:
1628:
1625:
1623:
1620:
1618:
1615:
1613:
1610:
1608:
1605:
1603:
1600:
1598:
1595:
1591:
1588:
1586:
1583:
1582:
1581:
1578:
1576:
1573:
1569:
1566:
1565:
1564:
1561:
1557:
1554:
1553:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1500:
1497:
1496:
1495:
1492:
1490:
1487:
1485:
1482:
1478:
1475:
1474:
1473:
1470:
1466:
1463:
1461:
1458:
1456:
1453:
1452:
1451:
1448:
1446:
1443:
1442:
1439:
1433:
1430:
1428:
1425:
1423:
1420:
1419:
1416:
1411:
1410:Ancient Egypt
1404:
1399:
1397:
1392:
1390:
1385:
1384:
1381:
1373:
1372:
1367:
1363:
1359:
1353:
1349:
1345:
1344:
1340:
1335:
1331:
1330:
1319:
1313:
1308:
1302:
1296:
1291:
1282:
1280:
1278:
1276:
1269:
1265:
1259:
1257:
1255:
1253:
1251:
1249:
1247:
1245:
1236:
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1089:Corinna Rossi
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105:Third Dynasty
102:
99:
95:
91:
90:Palermo stone
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65:
63:
59:
54:
52:
48:
44:
40:
39:Ancient Egypt
36:
32:
26:
21:
1450:Architecture
1370:
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1327:Bibliography
1307:
1290:
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1050:. Retrieved
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404:
363:
189:
167:
148:
124:
116:Step Pyramid
109:
83:
71:
55:
30:
29:
1788:WikiProject
1602:Mathematics
1563:Hieroglyphs
1477:Portraiture
1445:Agriculture
1432:Main topics
1059:disappear."
653:Hemisphere:
460:Rectangles:
452:= base and
179:New Kingdom
144:Tutankhamun
1803:Categories
1718:Egyptology
1686:Technology
1649:Philosophy
1597:Literature
1489:Chronology
1234:0262070456
1205:0900416327
1180:0900416351
1155:0521448492
1115:0486264858
1020:References
628:3.16049...
406:Triangles:
394:hemisphere
377:3.14159...
257:rectangles
94:Nile River
51:Nile river
33:refers to
1622:Mythology
1546:Geography
1536:Dynasties
1484:Astronomy
1341:Vol. III:
1052:9 January
507:−
371:π
47:surveying
1758:Category
1679:District
1674:Capitals
1659:Religion
1642:Titulary
1632:Pharaohs
1612:Military
1607:Medicine
1590:Hieratic
1580:Language
1506:Clothing
1460:Obelisks
1368:(1865).
1336:(1999).
480:Circles:
207:triangle
183:Senenmut
138:tomb of
43:geometry
41:. Their
35:geometry
1778:Outline
1768:Commons
1728:Museums
1664:Scribes
1654:Pottery
1585:Demotic
1575:History
1526:Cuisine
1455:Revival
1227:. MIT.
700:Volumes
660:Volumes
136:Saqqara
132:Lepsius
120:Saqqara
98:pharaoh
1627:People
1494:Cities
1412:topics
1354:
1266:
1231:
1202:
1177:
1152:
1112:
1043:
916:MMP 14
892:
886:
844:
815:
756:
720:RMP 41
709:Source
706:Object
448:where
297:circle
199:Source
196:Object
174:Thebes
155:Thebes
1691:Trade
1669:Sites
1617:Music
1531:Dance
1465:Pylon
1427:Index
1318:LV.4
1301:IV.3
1008:seked
1004:seked
998:Seked
190:Areas
170:Menna
153:) in
127:cubit
1637:List
1556:List
1499:List
1352:ISBN
1264:ISBN
1229:ISBN
1200:ISBN
1175:ISBN
1150:ISBN
1110:ISBN
1054:2020
1041:ISBN
140:Maya
110:The
101:Djer
68:Area
23:The
1472:Art
826:128
738:256
331:256
172:in
151:TT8
118:of
1805::
1346:.
1274:^
1243:^
1136:^
1096:^
1081:^
1066:^
1056:.
829:27
800:27
797:32
741:81
564:64
544:63
334:81
146:.
53:.
1402:e
1395:t
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1360:.
1237:.
1208:.
1183:.
1158:.
1118:.
1076:.
980:h
977:)
972:2
968:b
964:+
961:b
958:a
955:+
950:2
946:a
942:(
937:3
934:1
929:=
926:V
895:h
889:l
883:w
880:=
877:V
847:h
839:2
835:r
821:=
818:h
810:2
806:d
792:=
789:V
759:h
751:2
747:r
733:=
730:V
625:=
620:2
616:)
610:9
607:8
602:(
599:4
575:2
571:8
567:=
541:=
538:)
535:3
532:(
529:)
526:3
523:(
518:2
515:1
510:4
502:2
498:9
454:h
450:b
436:h
433:b
428:2
425:1
420:=
417:A
374:=
347:2
343:d
339:)
326:(
321:4
318:1
313:=
310:A
279:h
276:b
273:=
270:A
239:h
236:b
231:2
228:1
223:=
220:A
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