Brahmagupta's cyclic-quadrilateral formula: Heron, one dimension further, 1,200 years before Europe

# Brahmagupta's cyclic-quadrilateral formula: Heron, one dimension further, 1,200 years before Europe

In 60 CE — give or take a few decades — Heron of Alexandria wrote down a single formula that gives the area of any triangle from its three sides alone:

$$K = \sqrt{s(s-a)(s-b)(s-c)}$$

where $s = (a+b+c)/2$ is the semi-perimeter. Beautiful, symmetric, needs no angle, no height, no diagram. Plug in the sides, get the area.

In 628 CE, in Bhinmal in what is now Rajasthan, Brahmagupta wrote down the *next* formula in the sequence — the four-sided version:

$$K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$$

where $s = (a+b+c+d)/2$. Same shape. One extra factor. The four-sided analog of Heron.

There's a footnote that matters, and we'll get to it. But first, the shape of the achievement.

What the verse says

Brahmagupta's verse — *Brāhmasphuṭasiddhānta* XII.21 — is terse, in the compressed Sanskrit style of *sūtra* writing. Bhāskara II, 522 years later, preserved the algorithm verbatim in his more widely-read *Lilavati*:

> Half the sum of all the sides is set down in four places; and the > sides are severally subtracted. > > — *Lilavati* §167, restating *Brahmasphutasiddhānta* XII.21, trans. > H. T. Colebrooke (1817)

That is steps 1-2-3 of the algorithm: take half the perimeter, write it down four times, subtract each of the four sides from each copy. The verse's next sentence (which has an OCR-broken word in our cached scan so we don't quote it directly) gives steps 4-5: multiply the four remainders, take the square root, get the area.

In modern algebra, that procedure is exactly $K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$.

The constraint Brahmagupta didn't quite name

Here is the footnote that matters: Brahmagupta's formula is **exact only for cyclic quadrilaterals** — four-sided figures whose four vertices all lie on a single circle.

For an arbitrary quadrilateral, the same formula gives the *maximum* area achievable with those four side lengths, attained when (and only when) the quadrilateral can be inscribed in a circle.

Brahmagupta's verse doesn't state this restriction. Bhāskara II, six centuries later, noticed something was off and hedged the result by calling it "inexact in the quadrilateral, but pronounced exact in the triangle" — he saw the formula didn't match measured areas for arbitrary quadrilaterals, but he couldn't articulate cleanly why.

The cyclic-quadrilateral interpretation — that the formula is exactly right under exactly that geometric condition — was clarified by later Sanskrit commentators and made fully rigorous in modern European geometry in the 19th century.

So the honest framing is: Brahmagupta wrote down a true and useful formula, and the deep geometric reason it works (only when the quadrilateral inscribes in a circle) was understood layer by layer over the next 1,200 years. The result is correct; the necessary condition was implicit in the verse and made explicit only later.

What Europe had

For most of the period between Brahmagupta and Strehlke, European geometry had only Heron's triangle formula. The standard medieval European approach to quadrilateral area was: split into two triangles along a diagonal, apply Heron twice, add. That works, but it requires *knowing the diagonal*, which is exactly the unknown you don't have when you're given only the sides.

The first published European derivation of the cyclic-quadrilateral formula from sides alone is **Carl Friedrich Strehlke's 1842 paper** *Zwei neue Sätze über das ebene und sphärische Viereck* ("Two new propositions on the plane and spherical quadrilateral"), published in Crelle's *Journal für die reine und angewandte Mathematik*. That is 1,214 years after Brahmagupta's verse.

There is no evidence Strehlke had access to Brahmagupta — Colebrooke's 1817 English translation existed by then, but European geometers weren't generally reading Sanskrit mathematical literature for working results. The 19th-century rediscovery and Brahmagupta's original appear to be independent.

Brahmagupta and Heron

A natural question: did Brahmagupta know Heron's triangle formula?

The honest answer: probably not directly, and possibly not at all.

Heron's *Metrica* — the surviving Greek text where the triangle area formula appears — has a complicated transmission history. It was lost to medieval Europe and only rediscovered in the late 19th century in a single Greek manuscript at Constantinople. There's no clear evidence the Heron formula reached India in any tradition Brahmagupta would have had access to. Brahmagupta's verse doesn't cite Heron, doesn't echo Greek phrasing, doesn't use the Heron-style semi-perimeter as its own variable — it operates from first principles within the Sanskrit *śulba*-and-*ganita* geometric tradition.

The two formulas are best read as *independent* derivations of related results from related geometric intuitions, in two mathematical traditions that had limited contact during the late-classical period they both productively occupied.

What this leaves us with

Two formulas. Two traditions. One missing dimension.

Heron, c. 60 CE, gives the three-sided case. Brahmagupta, 628 CE, gives the four-sided cyclic case. Strehlke, 1842, rediscovers Brahmagupta. The general formula for any cyclic *n*-sided polygon exists today (Robbins, 1994), and the pentagon and hexagon cases involve algebra of considerably higher degree — but the quadrilateral is the last case with a clean four-term form.

What Brahmagupta wrote down in 628 CE is the cleanest geometric result of its kind. Half the perimeter; subtract each side; multiply; take the square root. That is the area of any cyclic quadrilateral, exactly. The formula has not been improved. It has only been re-derived.

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Sources

- [Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhāscara, H. T. Colebrooke trans., 1817](https://archive.org/details/1817-henry-colebrooke-algebra-with-arithmetic-and-mensuration-from-the-sanskrit-of-br) — Brahmasphutasiddhanta XII.21 and Lilavati §167 cited above. - Plofker, K. (2009). *Mathematics in India*. Princeton University Press, ch. 5. — places the cyclic-quadrilateral formula in the arc of Indian mensurational geometry. - Strehlke, C. F. (1842). "Zwei neue Sätze über das ebene und sphärische Viereck und über die Auflösung der Gleichungen vom vierten Grade." *Journal für die reine und angewandte Mathematik* 24, 240. — the European rediscovery. - Robbins, D. P. (1994). "Areas of polygons inscribed in a circle." *Discrete & Computational Geometry* 12, 223-236. — the general *n*-sided extension of Heron-Brahmagupta.

Related claims

- [Brahmagupta on the arithmetic of zero](/c/d10352e6-dc8c-58ee-bbbf-9b1489efdc9e) — the same author, the same treatise, six chapters earlier.