In a triangular pyramid, the perimeters of all its faces are equal. Find the area of the total surface of this pyramid if the area of one of its faces is $S$.

#
Suppose the area of the face $A B C$ is $S$ (fig.). Let's write that the perimeters of all faces are equal:

$$
A K+K B+A B=A B+A C+C B,(1)
$$

$$
A K+K C+A C=K C+C B+K B,(2)
$$

$$
A K+K B+A B=K B+B C+K C .
$$

By adding equalities (1) and (2) term by term, we find that $A K=B C$. Substituting $B C$ for $A K$ in equality (3), we find that $A B=K C$. Then from (1), $A C=K B$. From this, it easily follows that all faces of the tetrahedron are equal to each other, and the area of its surface is $4 S$.

![](https://cdn.mathpix.com/cropped/2024_05_06_bca77ef9e87e9e47ba11g-22.jpg?height=569&width=652&top_left_y=1528&top_left_x=703)

Answer

$4 S$.

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