A tetrahedron is a three-dimensonal figure where each side is an equilateral triangle. Therefore, each angle in the triangle is \(\displaystyle 60^{\circ}\).

In the figure, we know the value of the side \(\displaystyle (2\: m)\) and the value of the base \(\displaystyle (1\: m)\). Since dividing the triangle by half creates a \(\displaystyle 30^{\circ}-60^{\circ}-90^{\circ}\) triangle, we know the value of \(\displaystyle h\) must be \(\displaystyle \sqrt{3}\: m\).

Therefore, the area of one side of the tetrahedron is:

\(\displaystyle A = (b)(h) = (1\: m)(\sqrt{3}\: m) = \sqrt{3}\: m^2\)

Since there are four sides of a tetrahedron, the surface area is:

\(\displaystyle SA = 4 (b)(h) = 4\sqrt{3}\: m^2\)

The area of one face of the triangle can be found either through trigonometry or the Pythagorean Theorem.

Since all the sides of the triangle are \(\displaystyle 1\), the height is then\(\displaystyle \frac{\sqrt{3}}{2}\), so the area of each face is:

\(\displaystyle \frac{bh}{2}=\frac{1}{2} \cdot 1 \cdot \frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}\)

There are four faces, so the area of the tetrahedron is:

 \(\displaystyle 4 \cdot \frac{bh}{2}=\sqrt3\)

Use the following formula to find the surface area of a regular tetrahedron.

\(\displaystyle \text{Surface Area}=\sqrt3 (side^2)\)

Now, substitute in the value of the side length into the equation.

\(\displaystyle \text{Surface Area}=\sqrt3(4^2)=16\sqrt3\)

Use the following formula to find the surface area of a regular tetrahedron.

\(\displaystyle \text{Surface Area}=\sqrt3 (side^2)\)

Now, substitute in the value of the side length into the equation.

\(\displaystyle \text{Surface Area}=\sqrt3(12^2)=144\sqrt3\)

Use the following formula to find the surface area of a regular tetrahedron.

\(\displaystyle \text{Surface Area}=\sqrt3 (side^2)\)

Now, substitute in the value of the side length into the equation.

\(\displaystyle \text{Surface Area}=\sqrt3(5x)^2=25x^2\sqrt3\)

Use the following formula to find the surface area of a regular tetrahedron.

\(\displaystyle \text{Surface Area}=\sqrt3 (side^2)\)

Now, substitute in the value of the side length into the equation.

\(\displaystyle \text{Surface Area}=\sqrt3(3q)^2=9q^2\sqrt3\)

Use the following formula to find the surface area of a regular tetrahedron.

\(\displaystyle \text{Surface Area}=\sqrt3 (side^2)\)

Now, substitute in the value of the side length into the equation.

\(\displaystyle 36\sqrt3=\sqrt3(3a)^2\)

Now, solve for \(\displaystyle a\).

\(\displaystyle 9a^2=36\)

\(\displaystyle a^2=4\)

\(\displaystyle a=2\)

Use the following formula to find the surface area of a regular tetrahedron.

\(\displaystyle \text{Surface Area}=\sqrt3 (side^2)\)

Now, substitute in the value of the side length into the equation.

\(\displaystyle 54=\sqrt3(6x)^2\)

Solve for \(\displaystyle x\).

\(\displaystyle 36x^2\sqrt3=54\)

\(\displaystyle x^2\sqrt3=1.5\)

\(\displaystyle x^2=\frac{1.5}{\sqrt3}=0.866\)

\(\displaystyle x=\sqrt{0.866}=0.9\)

Use the following formula to find the surface area of a regular tetrahedron.

\(\displaystyle \text{Surface Area}=\sqrt3 (side^2)\)

Now, substitute in the value of the side length into the equation.

\(\displaystyle 64=\sqrt3(4x)^2\)

\(\displaystyle 16x^2=\frac{64}{\sqrt3}\)

\(\displaystyle x^2=\frac{64}{16\sqrt3}=2.309\)

\(\displaystyle x=1.5\)

Use the following formula to find the surface area of a regular tetrahedron.

\(\displaystyle \text{Surface Area}=\sqrt3 (side^2)\)

Now, substitute in the value of the side length into the equation and solve for \(\displaystyle x\).

\(\displaystyle 100\sqrt3=\sqrt3(x-1)^2\)

\(\displaystyle (x-1)^2=100\)

\(\displaystyle x^2-2x+1=100\)

\(\displaystyle x^2-2x-99=0\)

\(\displaystyle (x-11)(x+9)=0\)

\(\displaystyle x=11, x=-9\)

Since we are dealing with a 3-dimensional shape, only \(\displaystyle x=11\) is valid.