The surface area is the area of all of the faces of the tetrahedron. To begin, we must find the area of one of the faces. Because a tetrahedron is made up of triangles, we simply plug the given values for base and height into the formula for the area of a triangle:

Therefore, the area of one of the faces of the tetrahedron is . However, because a tetrahedron has 4 faces, in order to find the surface area, we must multiply this number by 4:

Therefore, the surface area of the tetrahedron is .

If this is a regular tetrahedron, then all four triangles are equilateral triangles. 

If the slant height is , then that equates to the height of any of the triangles being .

In order to solve for the surface area, we can use the formula

where  in this case is the measure of the edge.

The problem has not given the edge; however, it has provided information that will allow us to solve for the edge and therefore the surface area. 

Picture an equilateral triangle with a height .

Drawing in the height will divide the equilateral triangle into two 30/60/90 right triangles. Because this is an equilateral triangle, we can deduce that finding the measure of the hypotenuse will suffice to solve for the edge length (). 

In order to solve for the hypotenuse of one of the right triangles, either trig functions or the rules of the special 30/60/90 triangle can be used. 

Using trig functions, one option is using .

Rearranging the equation to solve for , 

Now that  has been solved for, it can be substituted into the surface area equation.

The problem is essentially asking us to go from a three-dimensional measurement to a two-dimensional one. In order to approach the problem, it's helpful to see how volume and surface area are related. 

This can be done by comparing the formulas for surface area and volume:

We can see that both calculation revolve around the edge length.

That means, if we can solve for  (edge length) using volume, we can solve for the surface area. 

Now that we know , we can substitute this value in for the surface area formula:

Write the volume equation for a tetrahedron.

In this formula,  stands for the tetrahedron's volume and  stands for the length of one of its edges.

Substitute the given edge length and solve.

Rationalize the denominator.

Write the equation to find the volume of a tetrahedron.

Substitute the side length and solve for the volume.

Rationalize the denominator.

The volume of a tetrahedron can be solved for by using the equation:

where  is the measurement of the edge of the tetrahedron. 

This problem can be quickly solved by substituting 6 in for . 

Solve the equaiton for the surface area of a sphere for the radius and plug in the values:

Recall that the equation for the volume of a sphere is:

For our data, we know:

Solve for . First, multiply both sides by :

Now, divide out the :

Using your calculator, you can solve for . Remember, if need be, you can raise  to the power of  if your calculator does not have a variable-root button.

This gives you:

If you get something like , just round up. This is a rounding issue with some calculators.

Recall that the equation for the volume of a sphere is:

For our data, we know:

Solve for . Begin by dividing out the  from both sides:

Next, multiply both sides by :

Using your calculator, solve for . Recall that you can always use the  power if you don't have a variable-root button.  

You should get:

  If you get , just round up to . This is a general rounding problem with calculators. Since you are looking for the diameter, you must double this to .

Recall that the surface area of a sphere is found by the equation:

For our data, this means:

Solve for . First, divide by :

Take the square root of both sides: