Brahmagupta · Mathematics
Brahmagupta’s celebrated formula for the area of a four-sided figure from its sides alone - the generalisation of Heron’s triangle formula - together with his deep results on cyclic quadrilaterals, and the subtle truth that the formula is exact only when the corners lie on a circle.
Long before Brahmagupta, Heron of Alexandria had given a lovely formula for the area of a triangle from the lengths of its three sides alone: take the semiperimeter (half the sum of the sides), subtract each side from it, multiply the three results together with the semiperimeter, and take the square root. Brahmagupta’s Brahmasphutasiddhanta reaches further, to the four-sided figure. His rule: for a quadrilateral with sides a, b, c, d and semiperimeter s, the area is the square root of (s - a)(s - b)(s - c)(s - d). It is the natural big brother of Heron’s formula - indeed Heron’s is the special case you get by letting one side shrink to nothing, so the quadrilateral collapses into a triangle. In one clean expression Brahmagupta extended the measurement of straight-sided figures from three sides to four.
Written out, the formula has a memorable symmetry. Let the four sides be a, b, c, d, and let s be the semiperimeter, s = (a + b + c + d) / 2. Then the area of the cyclic quadrilateral is the square root of the product (s - a)(s - b)(s - c)(s - d). Every side is treated alike; each is subtracted once from the semiperimeter, the four differences are multiplied, and the square root is taken. Set d = 0 and the fourth factor becomes (s - 0) = s, while the figure becomes a triangle, and the expression turns into Heron’s s(s - a)(s - b)(s - c) under the root - a satisfying confirmation that the four-sided rule genuinely contains the three-sided one. Brahmagupta stated it as a rule of mensuration, alongside a coarser, approximate area for rough work, and distinguished the two: a ‘gross’ estimate and this ‘exact’ result.
Brahmagupta’s work on quadrilaterals goes well beyond the area rule. He gave a separate ‘gross’ (approximate) area, the product of the half-sums of opposite sides, useful for rough estimates, and set it against the ‘exact’ area above - a striking distinction that suggests he knew the two answers could differ and that the exact one was special. He also gave formulas for the diagonals of a cyclic quadrilateral in terms of its sides, and a remarkable result now called Brahmagupta’s theorem: in a cyclic quadrilateral whose diagonals cross at right angles, the perpendicular dropped from that crossing to a side, extended, bisects the opposite side. Most impressive of all, he showed how to construct cyclic quadrilaterals whose sides, diagonals, and area are all whole numbers - the ‘Brahmagupta quadrilaterals’ - a feat of number theory dressed as geometry.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned Brahmagupta’s formula for the area of a cyclic quadrilateral from its four sides (the semiperimeter form that generalises Heron’s triangle formula), that it is exact only when the vertices lie on a circle, and that Brahmagupta also gave results on the diagonals and on rational ‘Brahmagupta’ figures. Explai…
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