2024 AMC 12A Problems/Problem 24

Problem

A $\textit{disphenoid}$ 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?

$\textbf{(A) }\sqrt{3}\qquad\textbf{(B) }3\sqrt{15}\qquad\textbf{(C) }15\qquad\textbf{(D) }15\sqrt{7}\qquad\textbf{(E) }24\sqrt{6}$

Solution 1 (Definition of disphenoid)

By definition, if a $\textit{disphenoid}$ has sides $x,y,z$ such that $x<y<z$ (since it is scalene), then we must have $x^2+y^2>z^2$ . Clearly the smallest triple of $(x,y,z)$ is $(4,5,6)$ . Then using Heron's formula gives us Surface area $= 4\sqrt{\frac{15}{2}(\frac{7}{2})(\frac{3}{2})(\frac{1}{2})}=\boxed{\textbf{(D) }15\sqrt{7}}$

2024 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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2024 AMC 12A Problems/Problem 24

Problem

Solution 1 (Definition of disphenoid)

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