1. f'he Mathematics Education SECT'ION B
V ol. VI I I, N o . 2 , J u n e 1 97 *
G L I M PS E S OF A N C IE N T IN D IAN tv|ATHEM ATICSNO. 10
fBrahrna$upta's Forrnulas for the Area and Dla$o-
nals of a Cycltc Gluadrllateral
D7R. C. Gupta Asristant
Professor Mathematics
of of
Birla Institute Techuology
P.O. Mesra,RANCHI.
(Received l5 APril 1974)
I fntroduction:
Let ABCD bea plane (convex) quadrilateral with sidesAB, BC,CD and DA equal
to a, b, c and d respectively. Let the figure be drawn in such a manner that we may
corrsider,according to the traditional terminoiogy, the side BC to be the base (bh[)
the side AD to be the face (mukha), and the sides AB and DCto be the flank sides (bhu-
jas or arms) of the quadrilateral.
Since a quadrilareral is not uniquely defined by its four sides, its shape and size
arc not fixed. So that, by merely specifying the four sides, the question of finding its
area does not arise. arise. That is why era'rrq Aryabhat II ( 950 A. D. ) in his cgrfir€ra
Mahi-siddhanta (-lvIS), XV, 70 saysr.
+rrinr*t fqqr sg{d qr.[+ si{ qil I
qlss] ${:
EEd qrsqfdrrqrs] fq{rs} st lleoll
Karna-jirZrreua vintr caturasre lambakar.nphalalr yadva /
Vaktu4r vErlchati gairako yo'sau mfirkhah pidaco va llTOll
'The mathematician who desires to tell the area or the altitude of a quadrilateral
without knowing a diagonal, is either a fool or a devil'.
Brahmagupta ( 628 A. D. )t in his ilr€EiuFsdrd Brahma-sphuta-siddbanta (- BSS)
has given two rules (see below) for finding the area of a quadrilateral in terms of its
fourgiven sides. One of the rules is forgetting a rough value of the area and the other
for an accurate (s[ksma) value. Now, Brahmagupta's forrnula for the accurate are of a qua-
drilateral gives the cxact value only when the quadrilateral 'is cyclic, although he has
not sptcified this condition, But the condition may be taken to be understood, especially when
we know (see below) that his expressions for the diagonals of the quadrilateral are also
true only whcn the figure is cyclic. otherwise the diagonals have remained undefined. In
fact, Brahmagupta docs speak of the circum circle (koqasplg-vltta) and the circum-radius

2. ' ,t Ht M AIIHEnAIIICB ED U OAT T oN
3+
which
( hldaya-rajju ) ot'rrianglc arrd quadrilateral in colule(ri.rn rvitlt some other rules2
are givcrr bltrvr-.e'rr lris rrrle frlr the alea and that fol the diagonals of thc ( cyclic )
qtradrilatcritl .
2 Rules for the Area :
' l' h" BSS, XtI, 2l V, r l. I I , p. Bl6) s t at t ' s
(Ti|$q ft<gg'wargxfter6a)rl<<era: I
qSqII ll Rt ll
glalirrtf ?Euaq gqlqqTilIicE
Stlrrlla-phalarlrtricatu r blruja-beltupratibrhu-yogadalaglfi tah /
B h rrjayogardha-catus a bhLrjorra-ghatatpada4r
tay sIkiamam //2 | //
' ' f he pr odu <t o l ' h a l f (h e s rrrn s l ' (th e two pai rs of ) the opposi te si desof a tri angl e
o
or a qrradrilateral, gives the gross area. Set down half the sum of the sides irr f<rrrr pla-
ces (and) diminish them by the (four) sides (respectively).The square-root of the prod-
d u rt ( of t he f our n u m l te rs ) i s th e a c c u ra tea re a' .
gross area : * (a * c ) . ! , t ( b- f d) (r)
accuratearea : l(-s-4t G4t Gt (r-l) (2)
where r: (a+1,+c*d12
The above ftrrmulas are stated to be applicable to the quatlrilateral as well as to
the triangle in which case we have to takr: the face d to be zero. Thus, in the case of a tria-
n q l t: r , f s ic lc s b . c, w e h a v e
a,
gr os sar c a :( b l z ). ( a -l c )l l (4)
accrrrateirrea
-/ s(s_a) ar_r) tr_r) (s)
$' lr t : ie s :( a f-b + c )l ' ) (6)
W e s eet hat i t d o e s n o t ma tte r m rrc h rv h etherthe fol mrrl a (l ) i s uscd for cycl i c or
o th e l c lr r adlr llt c la l s ,s i n c e , ; fre r a l l , i t i s s ta te d to be a rough one onl l ' . Fermul a (2) i s
krrown t'r give ex rct are.r r,nly in the ca:e of cyclic qu.adrilateral. However, the formula (5)
is applicable to erery triangle. Rut thc foln.irrla(4) has now an additional defect of not
vi e l d in, ; 21111iqe ( th o u g h ro u g l r ) v a l u e o f th e area ol ' a tri angl e, because w c may get
r
rl i fl rr r entr es r il s by rc g a rd i ri g e a c h o f th e s i d e sa, b, c to be ' base' i n turn,
A ny way , eq tri v a l e n tru l e s ; w h i c h y i e l d formul as (l ) and (2), have been gi ven by
sevelal sultsequer Indian rvriters with or without some additional comments.
t Some of
th e s ewill be not e, l n r,u ,.
Sri'lhara ( 4tq< ) in lris qrdtqfua Pfttiganita (:PG)a has reproduced, word by word,
BSS rule rvhich gives the formrrla (l). However. he adds the following remarks immedia-
tcly aftelv <luotingthe rules;
'But tl:is result (l) it true only f..r those figures in r,vhichthe difference between the
altitude and the flank sidesis small. In the caseof orher fisures the above result is far

3. R. ( j. GUPT A :.t5
removed from the truth; as for example, in the caseof the triangle having I3 for the tr.,"o
(flarrk) sidesand 24for the base, the gross area is 156, whereas the correct area is 60 (PG,
rules I l2-ll4)'.
An ancient commentator of the PG everr goes further and points out an interesting
theoretical defect ofthe rule (l), or (t) other than its grossness. FIe says (P. 160) that the
rule may yield a rough anlrwer'for thearea even in the caseof irnpossiblefigures, and gives
the example of a triangle of base 20 and flank sides 13 and 7. Since the sum of the two
sides is equal to the third (base), no tri:rngle is possible, but the formula (a) will give 100
for its gross area'.
The MS, XV, 69 (p.165) gives the BSS rrrle for the accurate area, but it is laid down
th e re for a t r ianglc o rrl y ' a n d n o t fo r a q u a d ri l ateral . B hi skara II (A .D . l l 50) i n hi s
Lilivati (eitmadl), rule 169, hasalso given the same rule but with the remark tltat it gives
evact area for a triangle and inexact (asphuta) for a quadrilateralG.
3. Sorne Historical and Other Rernarks:
The approximate fornrula (l) was used outside India much before thedate of Brahm-
agupta. The Babylonians of thc ancient.Mesopotamianvalley are stated to have used itin
finding the area of e quadrilateralT. The seme formula can be gathernd from the inscripti-
ons (about 100 B.C.) found on the 'Iemple of Horus at Edfn8. In this type of Egyptian
me n sur at ional m at h e n ra ti c s , tl re tri a n g l e s w e re regardede as cases of quadri l ateral s i n
which one sido (the face) is mad.: zero, just as what is met with in Brahnragupta.
The Chinese mathematical work Wu-t'sao Suan-ching( about 5th ot 6rh century )
applies the formula (l) for coniputing the area of a quadrangr,,ar field whose eastern,
western, southern, and northern sidesare given to be 35, 45r 25, and l5 pac6srespectivelylo.
The formula (5) Ibr the area of a triangle is generally ca"lledHeron's Formula, but,
according to same medival Arattic scholars, it was kuown everr to Archimedes (third century
B.C .;tt .
How Brahmagupta arrived at his formula (2), is difficult to say with certainty. For
an expostion of the attempted prool, of this formula, as given by GaneSa Daivajfra (qivrrien)
in his commentary (1545 A.D. l on tle Lilavati, a paper by M. G. Inamdar may be consul-
tedlr. The Ytrkti-bhnsl 1-y6, sixteenth century) also contains a proof of the same
formulal3.
4. Brahrnagupta's Expressions for the Diagonals:
The BSS, XIl,24 (Vol. III, p. 836) states
+<rtFragqqrti+agvaurfr;qqTfqilTqrtq r
alrrngwfagsratrq'l:nqit ca ftq} lrictl
Karnl ti rita-bhujaghfitaikyam-rrbhay:rthz'in1'onya-bh'jitam
gunayet /
logena bhujapratiblrda-vacllravoh k'rrnau pade visame ll2Sl
I
'The sums of the products of the sidesabout the diagonals be both divided by each
other; multiply (the quotients obtained) by the sum of the prodr cts of the opporite sides;

4. ED U OAT IO N
35 r IIT M ATHEM ATICE
'11'.
the lquare-root (of the results) are the two diagonalr (visame
I'hat is
AC:,/m (7)
BD:/M (B)
Brahmagupta's Sanskrit stanza, giving these-diagonals, by
has been quote-dra.vcrbatin
Bhrs126 It in"hii Lilivati witn the remark that 'although indeternrinate,the diagonals are
sought to es determinate by Brahmagupta and others'.
It may be noted that the, from (7) and (B), we immediately get ...(7)
A C . BD :a .c Ib .d ...(8)
which is called the Ptolemy's Theorem for cyclic quadrilateral after the famous Greek astro-
nomer of the secondcentuiy A. D, The YB (pp, 232-33), horlever, frrllcrvs the opposite
procedure of deriving (7) and (8) from (9) and someother relations.
t'most remarka'
Brahmagupta'.1 expressionsfor the diagonals are considered to be the
ble in Hindu gu"o-etry and solitary in its excellence" by a recent historian of mathematicslt'
The formnla [e) ir stitea to be rediscoveredro Europe by W. Snell (about l6l9 A.Dr).
in
Refercncee
I MS edited by S. Dvivedi, Fasciculur II, p. 165;Benares,19l0 (Braj Bhusan Das & Co.).
2 For a short decription of Brahmagupta's works, see R.C. Gupta, "Brahmagupt_a'sRule
for the Volume of Frustrrm like Solids", The Matl;enaticsEducation,Vol. VI., No'4
(D ec em ber 197 2 (,Sc e . B,p .l 1 7 .
3 BSS, XIl, 26-27.Edited by R.S. Sharma and his team, Vol. III, pp' B33-34; New Delhi,
1966 (Indian Inst. of Astronomical:rndSanskrit Research). All page refercncesto BSS
are according to this edition.
4 PG, rule ll2a. Editecl. with an ancient commentary, by K.S. Shukla, p. 156 of the text;
Lucknorv, iS5') (Lutknow IJuiversity). The editor has placed the author bsllvssn ti50
and 950 A.D., while sev.ral earlier scholarsplaced Sridhera before 850 A.D. (see l'G
introduction, p.xxxviii).
5 Ibid., translation, pp. 87-88.
6 The Lilivati, part II, p. 156.Edited bv D.V. Apte, Poona, 1937(Anandasrama Sanskrit
S er iesNo. 107) .
7 C.B. Boyer, A Hi;tory of Mathematics,p.42; New York, l968 (John Wiley).
B T.L. Heath, A ManualdCreek Mathematics, 77; Ncw York 1963 (Dover reprint).
p.
I Ibid., p. 7B.
l0 Y. Mikami, The Deuelopment MathcmaticsinCl.ina and Japan, p. 38; Nerv York, 196l
of
(Chelseareprint).
l l Boy er , op. c it . , p . 1 4 9 .
12 Nagpur Uaiuersitjt Journal, 1946, No.l I pp. 36-42.
13 YB (in Maiayalarn) part I, pp. 247-257. Edited by Rama Varma lVlaru Thampuran and
A.R. Aktrileswar Aiyar, Trichur, lg48 (Mangalodayarn Press).
l4 Lilsvati, part II, p. lB0.
l5 lroward Eves, An Introduction thc History of Mathematics,p.l}7; New York, 1969 (Holt,
to
Rinchart and Winston),
l6 D E. Smith, Historytof Mathemah'cs vol. II, p.287; New York, l95B (Dover reprint).