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Home » Second derivative confirms maximum (negative concavity). - Parker Core Knowledge

Second derivative confirms maximum (negative concavity). - Parker Core Knowledge

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Area of the Octagon:

📌 For a regular octagon with side length \( s = x \), the area \( A \) is given by:

📌 A_{\text{octagon}} = 2(1 + \sqrt{2})s^2 = 2(1 + \sqrt{2})x^2

📌 A_{\text{octagon}} = 2(1 + \sqrt{2})x^2

📌 The ratio of the area of the circle to the area of the octagon is:

📌 \text{Ratio} = \frac{A_{\text{circle}}}{A_{\text{octagon}}} = \frac{\pi x^2}{2(1 + \sqrt{2})x^2} = \frac{\pi}{2(1 + \sqrt{2})}

📌 Thus, the ratio of the area of the circle to the area of the octagon is \(\boxed{\frac{\pi}{2(1 + \sqrt{2})}}\).

📌 A computer scientist is designing an AI interface that uses equilateral triangular logic gates. Each gate has a side length of \( s \). If the side length is increased by 50%, what is the ratio of the new area to the original area of one triangular gate?

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