Ano he look a he nb. in he Moscow Ma hema ical
Papy us
Geo geM. Hollenback
ABSTRACT
This pape makes acase ha he nb. o “baske ” whose su ace a ea is calcula ed in P oblem 10 o he Moscow
Ma hema ical Papy us is asegmen o aci cle whose wo gi en dimensions – base o 9 and heigh o 4½–
iden i y i as asemici cula segmen . The a ea is ound by con igu ing he wo gi en dimensions as he sides
o a ec angle and hen educing he long side o he ec angle by wo successi e educ ions o   : 9 – = 8,
and 8 – =7   , he p oduc o 7   and 4½ yielding he a ea o he igu e, 32. The shape o he nb. has also
been cons ued by o he schola s as ahemisphe e and as he la e al a ea o asemicylinde , wi h plausible-
-sounding a gumen s mus e ed in a o o hose in e p e a ions. Those in e p e a ions, howe e , depend
upon ope a ions no o he wise a es ed in he Middle Egyp ian ma hema ical papy i. The in e p e a ion
pu o h he e is ha he nb. belongs o a amily o wo -dimensional plane igu es ha ing a la base called
a p ‑ , his dimension sho ened by a igu e -speci ic algo i hm and mul iplied oge he wi h agi en second
dimension o gi e he a ea o he igu e.
KEYWORDS
Moscow Ma hema ical Papy us– Egyp ian geome y– W. W. S u e– T. E ic Pee
ةيضايرلا وكسوم ةيدربب nb. ىلع ىرخأ ةرظن
كابنيلوه .م جروج
صخلملا
ءزج ىه ،ةيضايرلا وكسوم ةيدرب نم 10 ةلأسملا ىف اهحطس ةحاسم باسح مت ىتلا ،»ةلسلا« وأ nb. نأ ةقرولا هذه دكؤت
للاخ نم ةحاسملا ديدحت متي .ىرئاد فصن ءزج اهنأ ىلع - 4½ عافترلااو 9 ةدعاقلا - نافورعم نادعب اهددحي ةرئاد نم
:  9 – = 8
نييلاتتم نيضيفخت رادقمب ليطتسملا ىف لوطلأا علضلا ليلقت مث ،ليطتسمل علاضأك نيروكذملا نيدعبلا مادختسا
هنأ ىلع nb. لكش نورخآ ءاملع رسف دقو .32 ،لكشلا ةحاسم ىطعي 4½ ىف 7   برض لصاح ،8 – =7   كلذكو
تاريسفتلا كلت نإف ،كلذ عمو .تاريسفتلا هذه حلاصل ةلوقعم ودبت ججح ميدقت عم ،ةناوطسأ فصنل ةيبناج ةحاسم وأ ةرك فصن
ةقرولا هذه ىف مدقملا ريسفتلا .ىطسولا ةلودلا ىلإ دوعت ىتلا ةيرصملا ةيضايرلا تايدربلا ىف اهتابثإ متي مل تايلمع ىلع دمتعت
دعُبلا اذه صيلقت متي . p ‑ مساب فرعُت ةحطسم ةدعاق تاذ داعبلأا ةيئانث ةيوتسملا لاكشلأا نم ةلئاع ىلإ ىمتنت nb. نأ وه
.لكشلا ةحاسم داجيلإ ددحم ٍناث دعُب ىف دعبلا اذه برضُي مث ،لكشلاب ةصاخ ةيمزراوخ مادختساب
ةيحاتفملا تاملكلا
W. W. S u e – T. E ic Pee – ةيرصملا ةسدنهلا – ةيضايرلا وكسوم ةيدرب
PRAGUE EGYPTOLOGICAL STUDIES XXXIII/2024, 94–105
95GEORGEM. HOLLENBACK
The subjec o his piece is ama hema ical p oblem in which he p oblem sol e is asked o
calcula e he su ace a ea o some kind o baske -shaped geome ic igu e. Bu i u ns ou
o be a om as aigh o wa d ope a ion, howe e , because he e is adamaged wo d in he
ex and also apossible sc ibal omission o ameasu emen o one o he objec ’sdimensions.
These ambigui ies open up he possibili y o mo e han one in e p e a ion o jus exac ly
wha he objec is supposed o be. The p oposed shapes ha ha e gained he mos ac ion
a e ahemisphe e, asemici cle, and he la e al a ea o asemicylinde . This pape analyzes he
p oblem wi h a iew o asce aining which o hese shapes is mos likely o be he igh one.
E e since i s publica ion in 1930, P oblem 10 o he Moscow Ma hema ical Papy us (P.Mos-
cow 4676) has con inued o challenge i s in e p e e s. The p oblem deals wi h calcula ing he
su ace a ea o ageome ic igu e called anb. o “baske ”, which is desc ibed as being “hal an
i[�]”. Sugges ions as o exac ly wha he nb. is and how o econs uc i[�] ha e con inued o
appea in Egyp ological publica ions well in o he wen y - i s cen u y. This b ie s udy will
examine he a ious in e p e a ions o he objec , making acase ha one o he ea ly in e -
p e a ions, much o e looked and unde app ecia ed, is mos likely he co ec in e p e a ion.
Because he ex has len i sel o se e al di e en in e p e a ions, a echnical p e iew
o he p oposed shapes o he nb. and how o calcula e hei a eas would be in o de be o e
app oaching he ex . Hemisphe e: A ea = diame e  × semici cum e ence. Semicylinde (hal
he la e al a ea o acylinde di ided leng hwise): A ea = heigh  × semici cula a c. Semici cle:
A ea = adius × hal a c. The espec i e nume ical alues in each pai o dimensions a e 4½
and 7  , which yield ap oduc o 32. In addi ion o hese shapes, he e ha e been couple o
o he p oposals ha ha en’ gained as much o a ollowing (see ig. 4).
The e a e wo lineages o in e p e a ion o he p oblem, one based on he ex as i s ands,
which will be discussed i s , he o he based on an emended ex in which wha was hough
o be an omi ed nume ical dimension o he nb. is es o ed in line 2. The p oblem was o igi-
nally published by S u e (1930: 157–168, Ta el IV), and he ollowing is he English ansla ion
om he Ge man by Pee (1931: 100); in e linea ansli e a ion p o ided by he p esen w i e
(c . igs. 1–3):
1) Fo m o wo king ou abaske .
p n i . nb.
2) I hey men ion o you abaske wi h amou h
mi Dd n=k nb. m p ‑
3) o 4½ in p ese a ion.
4½ m aD HA
4) Le me know i s su ace.
di=k x=i AH. =s i i.x =k
5) Take anin h o 9, since he baske
i i=k
n 9 H ‑n i nb.
6) is hal an egg; esul 1.
gs pw n i[�] xp .x 1
7) Take he emainde , namely 8.
i i.x =k i i=k DA. m 8
96 PRAGUE EGYPTOLOGICAL STUDIES XXXIII/2024
Fig. 1 Lines 1–6 (pho o a e S u e 1930: Ta el IV)
Fig. 2 Lines 7–11 (pho o a e S u e 1930: Ta el IV)
Fig. 3 Lines 12–14 (pho o a e S u e 1930: Ta el IV)
97GEORGEM. HOLLENBACK
8) Take anin h o 8;
i i.x =k
n 8
9) esul ⅔ + ⅙ + ¹⁄₁₈. Take
xp .x ⅔ ⅙ ¹⁄₁₈ i i.x =k
10) he emainde o hese 8 a e ( he sub ac ion o )
i i=k DA. n. pA 8 ‑sA
11) his ⅔ + ⅙ + ¹⁄₁₈; esul 7   .
pA ⅔ ⅙ ¹⁄₁₈ xp .x 7
12) Recon wi h 7    4½ imes;
i i.x =k i i=k 7 zp 4½
13) esul 32. Behold, ha is i s su ace.
xp .x 32 mk AH =s pw
14) You ha e ound igh ly.
gmi=k n
He e he nb. is aken as ahemisphe e wi h adiame e o 4½ and acalcula ed semi-
-ci cum e ence o 7   , hose measu emen s mul iplied oge he o yield he a ea o i s cu ed
su ace, 32. Some o he Egyp ian e minology equi es commen . Adesc ip ion o he mou h
o he nb. in lines 2–3 eads nb. m p ‑ | 4 ½ m aD, S u e eading aD as he in ini i e o he e b
“ o be whole” (1930: 162–163); in o he wo ds, he mou h o p ‑ o he nb. is “whole” o “p e-
se ed”, i.e. he g ea ci cle o ahemisphe e and no some smalle ci cle belonging o alesse
segmen o asphe e. Bu p ‑ – li e ally “ on o he mou h”– is also used in Middle Egyp ian
ma hema ical papy i o e e o he bases o wo -dimensional plane igu es such as iangles
and apeziums, a ac no los on o he in e p e e s.1 In line 6, he nb. is desc ibed as being
“hal an i[�]”, he damaged wo d econs uc ed by S u e as in , li e ally “s one”, bu su ixed
wi h he egg sign o gi e i he meaning o “egg” o “eggshell”
(S u e 1930: 163–166).
The inding o he semici cum e ence o he nb. equi es some discussion. I aci cle is
enclosed in asqua e whose sides a e equal o he diame e o he ci cle, he e exis s a a io
o he a ea o he ci cle o he a ea o he squa e, aci cle - o -squa e a ea a io. Now he e is
an inhe en p opo ionali y in hese igu es such ha he ci cle - o -squa e pe ime e a io–
he a io o he ci cum e ence o he ci cle o he pe ime e o he squa e– is also he same
as he ci cle - o -squa e a ea a io.
2
P oblem 48 o he Rhind Ma hema ical Papy us depic s
a9 × 9 squa e wi h an a ea o 81 in which is enclosed aci cle (o apolygonal ep esen a ion o
aci cle) wi h adiame e o 9 and an a ea o 64, gi ing aci cle - o -squa e a ea a io o 64/81. I
he ancien Egyp ians we e cognisan o he equi alence o he a ea a io and he pe ime e
a io, hey would ha e been able o calcula e ci cum e ence by aking 64/81 o he pe ime e
o he squa e, which is equal o 4 imes he ci cle’sdiame e . Since he ancien Egyp ians didn’
1 Al hough mode n ma hema ical con en ion shows geome ic igu es such as iangles and ape-
ziums s anding on la bases, d awings o hese igu es in Middle Egyp ian papy i show ela i ely
all igu es ipped sideways, hei ela i ely na ow bases mo e o less e ically o ien ed. In ha
posi ion, he sides o he igu e esemble jaws, and he base esembles he “ on o he mou h”.
In he case o ahemisphe e, he “mou h” is qui e ob ious. Fo amilia i y’ssake, he plane igu es
depic ed in upcoming ig. 2 a e shown “s anding” on hei “bases”.
2 This can easily be con i med wi h pencil, pape , calcula o , and he π app oxima ion o one’schoice.
98 PRAGUE EGYPTOLOGICAL STUDIES XXXIII/2024
use ac ions such as 64/81 bu a he wo ked wi h uni ac ions, hey would ha e made wo
successi e educ ions o he squa e’spe ime e by o ob ain he same esul as i by aking
64/81 o he squa e’spe ime e . In o de o calcula e he semici cum e ence o aci cle ha ing
adiame e o 4½, he diame e would be doubled o ob ain he imagina y squa e’ssemipe im-
e e o 9– ha would be whe e he 9 comes om in line 5– and hen he 9 would be educed
by wo successi e educ ions o o ob ain he semici cum e ence o 7   . This is he iew
a icula ed by S u e (1930: 178).
Pee ound he g amma ical s uc u e o nb. m p ‑ | 4 ½ m aD p oblema ic because is
ypically used o in oduce he second o wo gi en dimensions. He esol ed he issue by
posi ing ano he dimension x o he nb. ha had inad e en ly been omi ed by he sc ibe:
nb. <n. x> m p ‑ | 4 ½ m aD (Pee 1931: 101). Thus he missing dimension was assigned o he
p ‑ , and he gi en dimension o 4½ was assigned o he aD, which Pee unde s ood o be he
name gi en o he second dimension. The nb. was x in i s p ‑ by 4½ in i s aD. Pee unde s ood
he i[�] in line 6 o be some kind o ound objec , simply calling i aci cle. By le ing he e-
s o ed dimension x be 9, he nb. could be unde s ood o be a wo -dimensional baske shape,
asemici cle (Pee 1931: 103):
1) Example o wo king asemici cle.
2) I hey say o you, Asemici cle <o 9> in diame e
3) by 4½ in heigh , p ay
4) le me know i s a ea. You a e o
5) ake anin h o 9, since asemici cle
6) is hal a[ci cle]
The e a e no changes o he es o he p oblem; 9 is educed by o 8, which is educed by
o 7   , which is mul iplied oge he wi h 4½ o ob ain 32.
Al hough Pee no es ha he 4½ co esponds o he adius o he semici cle and he 9 co -
esponds o he diame e , he doesn’ explain he con ex o he ma hema ical ope a ions
pe o med upon he diame e . Tha con ex can be supplied in he ligh o he p e iously
desc ibed ci cle - o -squa e a ios. In e ms o he pe ime e a io, he diame e o 9 is also
equal o one side o an imagina y 9 × 9 squa e; and pe o ming he ope a ions on aqua e
pe ime e o he squa e would yield he qua e ci cum e ence o i s enclosed ci cle, which
would equal he hal -a c o i s semici cle. Thus he 7   in he p oblem is he hal -a c o he
semici cle. Seidenbe g also unde s ands he calcula ion o he hal -a c o he semici cle as
a unc ion o pe ime e a io (Seidenbe g 1972: 196).
The a ea a io also a o ds asolu ion. I a9 × 9 squa e enclosing aci cle wi h adiame e
o 9 is bisec ed om side o side, he hal igu e would be asemici cle enclosed in a ec angle
measu ing 4½ × 9. Sho ening he leng h o he ec angle by wo successi e educ ions o
yields a ec angle leng h o 7   which when mul iplied oge he wi h he ec angle heigh
o 4½ gi es an a ea o 32. Pee had misgi ings abou he semici cle in e p e a ion, howe e ,
because i supplied wo dimensions when only one was necessa y (Pee 1931: 103, 106).
Pee o e ed asecond in e p e a ion which in ol ed le ing he missing dimension x be
4½ and aking he nb. as asemicylinde – hal he la e al a ea o acylinde di ided leng h
wise – whose heigh and diame e we e bo h 4½ (Pee 1931: 105):

99GEORGEM. HOLLENBACK
1) Example o wo king ou asemi -cylinde .
2) I hey say o you, asemi -cylinde <o 4½> in diame e
3) by 4½ in heigh ; p ay
4) le me know i s a ea. You a e o
5) ake anin h o 9, since asemi -cylinde
6) is hal o a[cylinde ].
And again, he e a e no changes o he es o he p oblem.
Pee en a i ely es o ed he damaged wo d i[�] in line 6 as ip
, which he ook as
ag ain measu e con aine wi h acylind ical shape, i s semicylind ical o m esembling he
cu e o ahemisphe ical baske when iewed endways (Pee 1931: 104–105). The ma hema ical
ope a ions in ol e he same applica ion o he pe ime e a io as in he calcula ion o he
semici cum e ence o he hemisphe e wi h adiame e o 4½; he e he calcula ed leng h o he
semici cula a c o 7   is mul iplied oge he he gi en heigh o 4½ o ob ain he a ea o 32.
Wi h bu acouple o excep ions, subsequen ea men s o P oblem 10 ha e ended o all
in o one o he h ee ca ego ies discussed abo e. Di e ences o opinion o e jus exac ly wha
he nb. is, wha he aD is, how lines 2–3 should be ead, and how i[�] should be es o ed ha e
kep he deba e going. He e is how hings s and a his w i ing ( ig. 4): hemisphe e (S u e
Fig. 4 Va ious in e p e a ions o he nb. ; he hemisphe e, semicylinde , and semici cle ha e been he
mos discussed
100 PRAGUE EGYPTOLOGICAL STUDIES XXXIII/2024
1930: 157–168; Gillings 1967; Gillings 1972: 194–201; Clage 1999: 234; Coope 2010; Michel 2012);
semici cle (Pee 1931: 104–105; Smeu 1970: 268; Seidenbe g 1972: 196; F ibe g 2005: 77–81) and
semicylinde (Pee 1931: 103–104; Ho mann 1996; Mia ello 2010; Mia ello 2013: 67–70). O he s
(e.g. Neugebaue 1969: 136–137) sugges ed he nb. migh be adomed g ana y wi h adiame e
o 4½ and an apex - o -ci cum e ence su ace dis ance o 4½, i s a ea being an app oxima-
ion a he han an accu a e calcula ion;3 Schwela (2011) cons ued he nb. as an ellipse wi h
dimensions o 4½ and 9.
I was s a ed ea ly on ha acase could be made o one o he ea lie in e p e a ions being
he igh one, and ha in e p e a ion is Pee ’ssemici cle. The i s poin o be add essed is
Pee ’sown misgi ings abou he semici cle because he p oblem gi es wo dimensions when
only one is necessa y, apoin echoed by o he in e p e e s as well.
4
The s umbling block he e is
he imposi ion o mode n sensibili ies upon ancien ma hema ics. In he mode n wo ld, e e y
schoolchild lea ns π 2, so when he a ea o asemici cle comes up, he e lexi e esponse is o
hink one hal π
2
. This wasn’ he case in ancien ma hema ics, howe e ; he Babylonians, o
example, had h ee o mulas o inding he a ea o asemici cle, none o hem amoun ing o
calcula ing he a ea o aci cle and aking one hal he esul (F ibe g 2005: 80).
The nex s ep is an applica ion o Occam’s azo o pa e away ap oblema ic aspec o all
he in e p e a ions based on pe ime e a io. The e is no in he en i e co pus o Middle
Egyp ian ma hema ical ex s asingle example o as aigh o wa d calcula ion o aci cle’sci -
cum e ence.5 Ye in in e p e a ions o his p oblem, he e a e impu ed calcula ions o semi-
-ci cum e ence and qua e ci cum e ence wi h no p eamble wha soe e – no ins uc ion
ha apa ial ci cum e ence needs o be calcula ed, no men ion o ci cum e ence a all, no
men ion ha he 9 in line 5 would ep esen adoubling o he diame e o 4½ in he case o
he hemisphe e and he semicylinde . The spu ious pe ime e a io business s ipped away,
he only emaining in e p e a ion is ha in ol ing a ea a io: he hal igu e o aci cle wi h
adiame e o 9 enclosed in a9 × 9 squa e is asemici cle enclosed in a4½ × 9 ec angle, and
ope a ing on he long side o he ec angle wi h wo successi e educ ions o sho ens he
leng h o he ec angle o 7   , lea ing he ec angle wi h an a ea o 32.
Be o e ela ing he me hod behind he a o emen ioned a ea a io calcula ion o he me hods
unde lying he a ea calcula ion o o he plane igu es, addi ional ema ks on e minology a e in
o de . In he Demo ic Ma hema ical Papy i, asegmen cu om aci cle by an insc ibed iangle
o squa e is called anby o “baske ” (Pa ke 1972: 44–48), and he baske glyph
in he wo d
nb. looks jus like one o he segmen s so depic ed; so pe haps he nb. may be mo e accu a ely
desc ibed as asegmen , which is de ined by wo dimensions. A e he wo dimensions o he
segmen a e disclosed, i hen becomes appa en ha he segmen is asemici cula segmen ,
equi ing apa icula me hod o a ea calcula ion. Recen schola ship has o e ed acon incing
econs uc ion o he damaged wo d i[�] as i n
(Michel 2012: 271), “sun” o “disk o sun”
3 The a ea is co ec o he la e al a ea o acone wi h abase o 4½ and aslan heigh o 4½ (see
Neugebaue 1969: 137).
4 I appa en ly escaped Pee ’sno ice ha asemici cle is aspecial case o aci cle segmen , which is
de ined by wo dimensions; o he in e p e e s, howe e , ha e no ed such (e.g Seidenbe g 1972: 195).
5 O he s ha e no ed his as well (e.g. Smeu 1970: 265); and “ he Egyp ians did no ‘calcula e’ he
ci cum e ence o aci cle” (Smeu 1970: 268, no. 76).
101GEORGEM. HOLLENBACK
(Faulkne 1962: 33), his s udy op ing o “disk” in he sense o a la , ci cula shape.6 The hi-
e a ic o aD isn’ pe ec ly clea , S u e eading i as being su ixed wi h he sc oll sign
and ha ing he meaning o he in ini i e o “ o be whole”, Pee eading i as being su ixed
wi h he canal sign
and pe haps ha ing he meaning o “ he s ip o land bo de ing on
he cul i a ion” (Pee 1931: 104, n. 3). This s udy eads he wo d as being su ixed wi h he canal
sign and ha ing he meaning o “edge” o “ma gin o cul i a ion” (Faulkne 1962: 51).
As p e iously men ioned, he bases o iangles and apeziums a e also called he p ‑ , hei
heigh e e ed o as he m y “bank” o “sho e” (Faulkne 1962: 112).7 Examples o calcula ing
he a ea o iangles and apeziums include P oblems 51 and 52 in he Rhind Ma hema ical
Papy us. The me hods a e hose s ill used oday; hal he base imes he heigh gi es he a ea
o he iangle, and hal he sum o he bases imes he heigh gi es he a ea o he apezium.
In hese wo examples, i is no ed by he sc ibe ha he ope a ions a e pe o med in o de
o ob ain he “ ec angle” o i d o he igu e (Pee 1923: 91, 94; Chace 1927: 92–93). This e e s
o he ac ha he a ea o he iangle is equal o he a ea o a ec angle whose dimensions
co espond o he heigh and hal base o he iangle, and ha he a ea o he apezium is
equal o he a ea o a ec angle whose dimensions co espond o he heigh and hal sum o
he bases o he apezium. I was subsequen ly disco e ed, howe e , ha wha was being
ead as i d o “ ec angle” should ha e ac ually been ead as i d ‑ mn o “hal a ec angle” (Galán
1990; Imhausen 2003: 251, 253). This men ion o hal ec angles implies whole ec angles whose
dimensions would hen co espond o he heigh and base o he iangle and o he heigh
and sum o he bases o he apezium- ec angles ha could be imagined as enclosing he
espec i e igu es.
This poin s o agene al me hod o se ing up he calcula ion o he a ea o plane igu es
ha ing a p ‑ . The pa icula igu e is isualised as being enclosed in a ec angle whose dimen-
sions co espond o he heigh and base ( p ‑ ) o he igu e. The base o he ec angle is hen
ope a ed upon in such away so as o lea e asmalle ec angle ha ing he same a ea as he
igu e, he heigh o he ec angle emaining unchanged ( ig. 5).8 In he case o he iangle,
he base o i s enclosing ec angle is sho ened by one hal . In he case o he apezium, he
base o i s enclosing ec angle is i s ex ended by he leng h o he uppe base, his sum o he
bases leng h hen sho ened by one hal . And in he case o he nb. , he base o i s enclosing
ec angle is sho ened by wo successi e educ ions o .
6 Michel he sel op s o “sun” in he sense o asphe ical body a he han “disk o sun” as a la ,
ci cula shape. This no e p esen s an oppo une ime o co ec amisconcep ion conce ning Neu-
gebaue ’spu po ed ake on i n. In no less han h ee ecen pape s on his p oblem, Neugebaue
is said o ha e econs uc ed i[�] as i n. In eali y, Neugebaue poin ed ou ha al hough some
eade s o he p oblem migh ake i[�] as i n, he himsel had his doub s, inally concluding: “So
is also die E gänzung i n mi Siche hei auszuschließen.” (Neugebaue 1969: 133) This s udy inds
Michel’sde ailed econs uc ion o he damaged wo d mo e pe suasi e han Neugebaue ’sobjec-
ions.
7 Middle Egyp ian geome y p oblems abound in ag a ian image y. The a ea o he nb. is i s AH. ,
li e ally “ ield” o “a able land” (Faulkne 1962: 4). I is no su p ising, hen, ha wo ds desc ibing
bounda ies o plo s o ac eage– “bank”, “sho e”, “edge”, “ma gin o cul i a ion”– would be used o
desc ibe he dimensions o geome ic igu es.
8 The apezium and iangle a e shown as igh -angle igu es as a isual aid o concep ualizing he
me hod desc ibed; he me hod gi es he same esul o isosceles and scalene igu es as well.
102 PRAGUE EGYPTOLOGICAL STUDIES XXXIII/2024
The c ucial beginning lines o he p oblem may he e o e be ende ed as ollows:
1) Me hod o calcula ing a segmen :
p n i . nb.
2) I said o you, asegmen <o 9> in base
mi Dd n=k nb. <n. 9> m p ‑
3) by 4 ½ in heigh , p ay
4 ½ m aD HA
4) le me know i s a ea.
di=k x=i AH. =s i i.x =k
5) Take o 9 because he segmen
i i=k
n 9 H ‑n i nb.
6) is hal a[ci cle]
gs pw n i[ n]
The pa adigma ic ci cle a ea p oblems in he Rhind Ma hema ical Papy us (41, 43, and 50)
all in ol e ci cles wi h adiame e o 9, he p oblem sol e ins uc ed o ake away o he
diame e o 9 as he i s s ep in calcula ing he a ea. The ins uc ions in line 5 abo e o “ ake
Fig. 5 Illus a ion o a me hod o a ea calcula ion in which a igu e is isualized as being enclosed in a
ec angle, he base o he ec angle ope a ed upon in such a way as o educe he a ea o he ec angle o he
a ea o he enclosed igu e