Visual proofs demonstrate the inequality πe < eπ using geometric constructions and areas under curves, suggesting new pedagogical methods.
We provide two simple and visual geometric proofs of the 𝜋 𝑒 < 𝑒 𝜋. These proofs are based on comparing areas under exponential curves and basic geometric shapes. Our approach builds on earlier visual arguments but introduces new geometric constructions that make the result more accessible for students and teachers. By highlighting these relationships visually, we hope to encourage further exploration and pedagogical use of similar inequalities involving transcendental numbers.
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Mandal et al. (2025) studied this question.
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