Area Approximation

To estimate the area of an equilateral triangle, square its side, and then take a third and a tenth of the square. Why is this a very close approximation?

Practica Geometriae, Leonardo of Pisa, 1220

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The answer is: for an equilateral triangle with side length \(a\), the exact area is \(\dfrac{\sqrt{3}}{4}a^2\), where \(\dfrac{\sqrt{3}}{4} \approx 0.43301\), while \(\dfrac{1}{3} + \dfrac{1}{10} = \dfrac{13}{30} \approx 0.43333\)

"Area Approximation," Convergence (August 2004)

Convergence