## Example Questions

### 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.

### Example Question #15 : Tetrahedrons

Given a regular tetrahedron with an edge of , what is the height (or diagonal)? The height is the line drawn from one vertex perpendicular to the opposite face.

None of the above.

Explanation:

The height of a regular tetrahedron can be derived from the formula

where  is the length of one edge.

Therefore, plugging in the side length of ,

.

### Example Question #16 : Tetrahedrons

Given a regular tetrahedron with an edge of , what is the height (or diagonal)? The height is the line drawn from one vertex perpendicular to the opposite face.

None of the above.

Explanation:

The height of a regular tetrahedron can be derived from the formula  where  is the length of one edge.

Plugging in  we can solve for .

### Example Question #17 : Tetrahedrons

Find the height of this regular tetrahedron:

Explanation:

The height of a regular tetrahedron can be found using the formula  where s is the length of the sides.

In this case, the sides have length , so we are multiplying .

We can simplify this by multiplying the numbers inside the radical:

, which simplifies to .

### Example Question #1 : How To Find The Surface Area Of A Tetrahedron

What is the surface area of the following tetrahedron? Assume the figure is a regular tetrahedron.

Explanation:

A tetrahedron is a three-dimensonal figure where each side is an equilateral triangle. Therefore, each angle in the triangle is .

In the figure, we know the value of the side  and the value of the base . Since dividing the triangle by half creates a  triangle, we know the value of  must be .

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

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

### Example Question #2 : How To Find The Surface Area Of A Tetrahedron

A regular tetrahedron has side lengths . What is the surface area of the described solid?

Explanation:

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 , the height is then, so the area of each face is:

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

### Example Question #3 : How To Find The Surface Area Of A Tetrahedron

Find the surface area of a regular tetrahedron with a side length of .