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

## Example Questions

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### Example Question #11 : Tetrahedrons

In order for the height of a regular tetrahedron to be one, what should the lengths of the sides be?

Explanation:

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

.

### Example Question #12 : Tetrahedrons

The volume of a regular tetrahedron is . Find the length of one side.

Explanation:

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:

.

### Example Question #13 : Tetrahedrons

A regular tetrahedron has surface area 1,000. Which of the following comes closest to the length of one edge?

Explanation:

A regular tetrahedron has six congruent edges and, as its faces, four congruent equilateral triangles. If we let  be the length of one edge, each face has as its area

;

the total surface area of the tetrahedron is therefore four times this, or

Set  and solve for :

Divide by :

Take the square root of both sides:

Of the given choices, 20 comes closest.

### Example Question #14 : Tetrahedrons

The above figure shows a triangular pyramid, or tetrahedron, on the three-dimensional coordinate axes. The tetrahedron has volume 1,000. Which of the following is closest to the value of ?

Explanation:

If we take the triangle on the -plane to be the base of the pyramid, this base has legs both of length ; its area is half the product of the lengths which is

Its height is the length of the side along the -axis, which is also of length

The volume of a pyramid is equal to one third the product of its height and the area of its base, so

Setting the volume  equal to 1,000, we can solve for :

Multiply both sides by 6:

Take the cube root of both sides:

The closest choice is 20.

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