# Advanced Geometry : How to find the length of an edge

## Example Questions

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### Example Question #1 : How To Find The Length Of An Edge

What is the length of one edge of a regular tetrahedron whose volume equals  ?

None of the above.

Explanation:

The formula for the volume of a tetrahedron is:

.

When  we have  .

Multiplying the left side by  gives us,

, or

Finally taking the third root of both sides yields

### Example Question #1 : How To Find The Length Of An Edge

A regular tetrahedron has a total surface area of . What is the combined length of all of its edges?

None of the above.

Explanation:

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

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

, or

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 .

Since the base  is the same as one edge of the tetrahedron, and a tetrahedron has six edges we multiply  to arrive at

### Example Question #3 : How To Find The Length Of An Edge

What is the length of one edge of a regular tetrahedron when the total surface area equals ?

None of the above.

Explanation:

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

.

### Example Question #4 : How To Find The Length Of An Edge

What is the length of one edge of a regular tetrahedron whose volume equals   ?

None of the above.

Explanation:

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.

### Example Question #5 : How To Find The Length Of An Edge

What would the length of one edge of a regular tetrahedron be if the area of one side was  ?

None of the above.

Explanation:

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

.

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.

### Example Question #44 : Solid Geometry

A regular tetrahedron has a surface area of . Each face of the tetrahedron has a height of . What is the length of the base of one of the faces?

Explanation:

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 .

### Example Question #1 : How To Find The Length Of An Edge Of A Tetrahedron

What is the length of an edge of a regular tetrahedron if its surface area is 156?

Explanation:

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.

### Example Question #2 : How To Find The Length Of An Edge

What is the length of a regular tetrahedron if one face has an area of 43.3 squared units and a slant height of ?

Cannot be determined

Explanation:

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:

### Example Question #47 : Solid Geometry

The volume of a regular tetrahedron is 94.8. What is the measurement of one of its edges?

Cannot be determined

Explanation:

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.

### Example Question #48 : Solid Geometry

A tetrahedron has a volume that is twice the surface area times the edge. What is the length of the edge? (In the answer choices,  represents edge.)