A regular tetrahedron has four faces of equal area made of equilateral triangles.

Therefore, we know that one face will be equal to,

  cm , or  cm.

 Since the surface of one face is an equilateral triangle, and we know that,

  , the problem can be expressed as:

In an equilateral triangle, the height  is equal to  so we can substitute for  like so:

Solving for  gives us the length of one edge.

However, we know that the edge of the tetrahedron is a positive number so

.

The formula for the volume of a tetrahedron is .  When  we have  .  

We simply solve for ...

.

Take the cube root of both sides to find the answer for a.

The area of one side is given as  .  The side of a regular tetrahedron is an equilateral triangle so area is determined by:

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In an equilateral triangle,   so we can substitute for  into the area formula:

.

Plugging in the value of the area which was given yields.

Solve for  will give us the length of an edge.

A regular tetrahedron has 4 triangular faces. The area of one of these faces is given by:

Because the surface area is the area of all 4 faces combined, in order to find the area for one of the faces only, we must divide the surface area by 4. We know that the surface area is , therefore:

Since we now have the area of one face, and we know the height of one face is , we can now plug these values into the original formula:

Therefore, the length of the base of one face is .

The only given information is the surface area of the regular tetrahedron.

This is a quick problem that can be easily solved for by using the formula for the surface area of a tetrahedron:

If we substitute in the given infomation, we are left with the edge being the only unknown. 

The problem provides the information for the slant height and the area of one of the equilateral triangle faces. 

The slant height merely refers to the height of this equilateral triangle. 

Therefore, if we're given the area of a triangle and it's height, we should be able to solve for it's base. The base in this case will equate to the measurement of the edge. It's helpful to remember that in this case, because all faces are equilateral triangles, the measure of one length will equate to the length of all other edges.

We can use the equation that will allow us to solve for the area of a triangle:

where  is base length and  is height.

Substituting in the information that's been provided, we get:

This becomes a quick problem by just utilizing the formula for the volume of a tetrahedron. 

Upon substituting the value for the volume into the formula, we are left with , which represents the edge length. 

The problem states that the volume is:

The point of the problem is to solve for the length of the edge. Becasuse there are no numbers, the final answer will be an expression. 

In order to solve for it, we will have to rearrange the formula for volume in terms of . 

The formula for the height of a regular tetrahedron is , where s is the length of the sides.

In this case we want h to be 1, so we need something that multiplies to 1 with .

We know that  , so then we know that

, which equals 1.

Therefore s should be

.

The formula for the volume of a regular tetrahedron is .

In this case we know that the volume, V, is , so we can plug that in to solve for s, the length of each edge:

[multiply both sides by ]

[evaluate and multiply]

[take the cube root of each side]

. 

We can simplify this by factoring 120 as the product of 8 times 15. Since the cube root of 8 is 2, we get:

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