WeChall - Math Pyramid

Challenge

A regular square pyramid where all 8 edges are equal in length (set to a). Find a formula for the volume (no more than 9 characters).

Solution

正四棱锥，所有 8 条棱等长（设为 a）。求体积公式（不大于 9 字符）。

底面正方形对角线 \(d = a\sqrt{2}\)，底面中心到顶点距离为 \(d/2 = a\sqrt{2}/2\)。

高 \(h = \sqrt{a^2 - (a\sqrt{2}/2)^2} = \sqrt{a^2 - a^2/2} = a/\sqrt{2}\)

体积 \(V = \frac{1}{3} \times a^2 \times h = \frac{1}{3} \times a^2 \times \frac{a}{\sqrt{2}} = \frac{a^3}{3\sqrt{2}}\)

化简到 9 字符以内：

\[ \frac{a^3}{3\sqrt{2}} = \frac{a^3}{\sqrt{18}} = 18^{-\frac{1}{2}} \cdot a^3 \]

提交 \(18^{-0.5}a^3\) 即可（9 字符，精确解）：

18^-.5a3