Solve a tetrahedron surface area formula for s

The surface area \( A \) of a regular tetrahedron (a tetrahedron with all four faces being equilateral triangles) is given by the formula:

\[
A = \sqrt{3} s^2
\]

where:
- \( A \) is the surface area,
- \( s \) is the length of a side.

To solve this formula for \( s \), follow these steps:

1. Start with the surface area formula:
   \[
   A = \sqrt{3} s^2
   \]

2. Divide both sides by \( \sqrt{3} \) to isolate \( s^2 \):
   \[
   \frac{A}{\sqrt{3}} = s^2
   \]

3. Take the square root of both sides to solve for \( s \):
   \[
   s = \sqrt{\frac{A}{\sqrt{3}}}
   \]

Thus, the side length \( s \) in terms of the surface area \( A \) is:

\[
s = \sqrt{\frac{A}{\sqrt{3}}}
\]

Let me know if you would like any further clarifications, and here are some related questions to explore:

1. What is the volume formula for a regular tetrahedron?
2. How is the height of a tetrahedron related to its side length?
3. How does the formula for surface area change for irregular tetrahedrons?
4. How would you compute the surface area if given the vertices in 3D space?
5. What role does the tetrahedron play in geometry and why is it significant?

**Tip**: For regular polyhedra like the tetrahedron, symmetry greatly simplifies calculations, especially for surface area and volume.

Discover how to solve for the side length of a regular tetrahedron when given its surface area. The problem covers key geometric concepts such as surface area and regular polyhedra, with a detailed step-by-step solution suitable for high school students.

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