07-16-2026, 11:06 AM
On Heron’s Formula for the Area of aPlane Triangle
Summary
In this article, Chris Sangwin revisits Heron’s formula, one of the most celebrated results in geometry, and explores why it remains such a remarkable achievement more than two thousand years after it was discovered. Rather than simply presenting the familiar formula $(A=\sqrt{s(s-a)(s-b)(s-c)})$, the paper examines its elegance, the ideas behind it, and the many different ways it can be derived using classical and modern mathematics. The author contrasts Heron’s formula with the standard “half base times height” approach, showing how Heron’s expression is especially powerful because it determines a triangle’s area using only its three side lengths, without requiring the height.
Along the way, the article connects geometry, algebra, trigonometry, and circle properties, illustrating how seemingly different mathematical concepts fit together. By presenting multiple proofs and historical insights, the paper demonstrates that Heron’s formula is far more than a computational shortcut—it is a beautiful example of the unity, creativity, and enduring appeal of mathematics, making it an ideal topic for both teaching and deeper mathematical exploration.
ARTICLE [PDF]
Summary
In this article, Chris Sangwin revisits Heron’s formula, one of the most celebrated results in geometry, and explores why it remains such a remarkable achievement more than two thousand years after it was discovered. Rather than simply presenting the familiar formula $(A=\sqrt{s(s-a)(s-b)(s-c)})$, the paper examines its elegance, the ideas behind it, and the many different ways it can be derived using classical and modern mathematics. The author contrasts Heron’s formula with the standard “half base times height” approach, showing how Heron’s expression is especially powerful because it determines a triangle’s area using only its three side lengths, without requiring the height.
Along the way, the article connects geometry, algebra, trigonometry, and circle properties, illustrating how seemingly different mathematical concepts fit together. By presenting multiple proofs and historical insights, the paper demonstrates that Heron’s formula is far more than a computational shortcut—it is a beautiful example of the unity, creativity, and enduring appeal of mathematics, making it an ideal topic for both teaching and deeper mathematical exploration.
ARTICLE [PDF]
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│ KONSTANTINOS MICHAILIDIS │
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