Question

等面四面体是指四个三角形面彼此全等的四面体。若一个等面四面体的各个面都是边长为整数的不等边三角形，则它的最小总表面积是多少？

A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?

$\sqrt3$

$3\sqrt{15}$

$15$

$15\sqrt7$

$24\sqrt6$

答案：D

知识点：立体几何海伦公式

难度评级：2520

解答：

等面四面体存在（可看作由一个长方体的面心构成的四面体）当且仅当公共面三角形是锐角三角形，总表面积为单个面的 $4$ 倍。 $u\lt v\lt w$ 。 $u\le3$ ， $w\ge v+1$ $u^2+v^2\le(v+1)^2$ $(u,v,w)=(3,4,5)$ $u\ge4$ 。

我们要找面积最小的锐角不等边整数三角形。候选 $60^\circ$ 和 $(u,v)\ne(4,5)$ 是钝角三角形， $\tfrac12uv\sin60^\circ$ 是直角三角形（会给出退化的平面图形），而 $(4,5,6)$ 是锐角三角形，因为 $4^2+5^2\gt6^2$ 。 $\ge\tfrac12(4)(6)\tfrac{\sqrt3}{2}=6\sqrt3$ ， $\tfrac{15\sqrt7}{4}$ ， $(4,5,6)$ 。

由海伦公式，半周长 $s=\tfrac{15}2$ ，面积为 $\sqrt{\tfrac{15}2\cdot\tfrac72\cdot\tfrac52\cdot\tfrac32}=\tfrac{15\sqrt7}{4}$ 。总表面积为 $4\cdot\tfrac{15\sqrt7}{4}=15\sqrt7$ 。

因此正确答案是 D。

A disphenoid exists (as the tetrahedron formed by the face-plane midpoints of a box) exactly when the common face triangle is acute, and its total surface area is $4$ times one face's area. Write the integer side lengths as $u\lt v\lt w.$ If $u\le3,$ then $w\ge v+1$ and $u^2+v^2\le(v+1)^2$ (with equality only for $(u,v,w)=(3,4,5)$ ), so the triangle is not acute. Thus $u\ge4.$

The triangle $(4,5,6)$ is acute because $4^2+5^2\gt6^2.$ It also has the least possible area. The largest angle of any acute triangle is at least $60^\circ,$ so a candidate with $(u,v)\ne(4,5)$ has area at least $\tfrac12uv\sin60^\circ$ $\ge\tfrac12(4)(6)\tfrac{\sqrt3}{2}=6\sqrt3,$ which is greater than $\tfrac{15\sqrt7}{4},$ the area of $(4,5,6).$

By Heron's formula with $s=\tfrac{15}2,$ that area is $\sqrt{\tfrac{15}2\cdot\tfrac72\cdot\tfrac52\cdot\tfrac32}=\tfrac{15\sqrt7}{4}.$ The total surface area is $4\cdot\tfrac{15\sqrt7}{4}=15\sqrt7.$

Thus, the correct answer is D.

2024 AMC 12A 第 24 题

答案：D

难度评级：2520

解答：

其他年份的第 24 题