# SSAT Upper Level Math : How to find the volume of a tetrahedron

## Example Questions

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

In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates .

Give its volume.

Explanation:

A tetrahedron is a triangular pyramid and can be looked at as such.

Three of the vertices -  - are on the -plane, and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:

Its base is 10 and its height is 18, so its area is

The fourth vertex is off the -plane; its perpendicular distance to the aforementioned face is its -coordinate, 8, so this is the height of the pyramid. The volume of the pyramid is

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

In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates .

What is the volume of this tetrahedron?

The correct answer is not among the other responses.

Explanation:

The tetrahedron looks like this:

is the origin and  are the other three points, which are fifteen units away from the origin on each of the three (perpendicular) axes.

This is a triangular pyramid, and we can consider  the base; its area is half the product of its legs, or

.

The volume of the tetrahedron is one third the product of its base and its height, the latter of which is 15. Therefore,

.

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

Above is the base of a triangular pyramid, which is equilateral. The height of the pyramid is equal to the perimeter of its base. In terms of , give the volume of the pyramid.

Explanation:

By the 30-60-90 Theorem, , or

is the midpoint of , so

The area of the triangular base is half the product of its base and its height:

The height of the pyramid is equal to the perimeter, so it will be three times , or

The volume of the pyramid is one third the product of this area and the height of the pyramid:

### Example Question #4 : How To Find The Volume Of A Tetrahedron

In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates

where

Give its volume in terms of .

Explanation:

The tetrahedron looks like this:

is the origin and  are the other three points.

This is a triangular pyramid, and we can consider  the base; its area is half the product of its legs, or

.

The volume of the tetrahedron is one third the product of its base and its height. Therefore,

After some rearrangement:

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

In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates

where

Give its volume in terms of .

Explanation:

The tetrahedron looks like this:

is the origin and  are the other three points, each of which lies along one of the three (mutually perpendicular) axes.

This is a triangular pyramid, and we can consider  the base; its area is half the product of its legs, or

.

The volume of the tetrahedron is one third the product of its base area  and its height . Therefore, the volume is

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

Find the volume of a regular tetrahedron that has a side length of .

Explanation:

Use the following formula to find the volume of a regular tetrahedron:

Now, plug in the given side length.

### Example Question #41 : Volume Of A Three Dimensional Figure

Find the volume of a regular tetrahedron that has a side length of .

Explanation:

Use the following formula to find the volume of a regular tetrahedron:

Now, plug in the given side length.

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

Find the volume of a regular tetrahedron with a side length of .

Explanation:

Use the following formula to find the volume of a regular tetrahedron:

Now, plug in the given side length.

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

Find the volume of a regular tetrahedron with side lengths of .

Explanation:

Use the following formula to find the volume of a regular tetrahedron:

Now, plug in the given side length.

### Example Question #10 : How To Find The Volume Of A Tetrahedron

Find the volume of a regular tetrahedron with side lengths of .