# GMAT Math : Calculating the volume of a tetrahedron

## Example Questions

### Example Question #1 : Calculating The Volume Of A Tetrahedron

A right triangular pyramid has as its base an equilateral triangle with sidelength 10. Its height is 15.

Give its volume.

Explanation:

The base of the triangle has an area that can be found using the formula for the area of an equilateral triangle, substituting :

Now, in the formula for the volume of a pyramid, substitute :

### Example Question #2 : Calculating The Volume Of A Tetrahedron

The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is one yard. Give its volume in cubic inches.

Explanation:

The perimeter of the base is one yard, or 36 inches; its sidelength - and, sunsequently, its height - are one-fourth of that, or 9 inches, and the area of the base is  square inches. The volume of a pyramid is one-third the product of its height and the area of its base, so substitute  in the following:

cubic inches.

### Example Question #3 : Calculating 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 60 and its height is 40, so its area is

The fourth vertex is off the -plane; its perpendicular distance to the aforementioned face is its -coordinate, 20, so this is the height of the pyramid. The volume of the pyramid is

.

### Example Question #4 : Calculating 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, whose distances away from the origin on each of the three (perpendicular) axes are shown.

This is a triangular pyramid, and we can consider  the (right triangular) 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 . Therefore,

.

### Example Question #5 : Calculating 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?

Explanation:

The tetrahedron looks like this:

is the origin and  are the other three points, which are 60 units away from the origin on each of the three (mutually perpendicular) axes.

This is a triangular pyramid, and we can consider  the (right triangular) 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 60. Therefore,

.

### Example Question #6 : Calculating The Volume Of A Tetrahedron

What is the volume of a right pyramid whose height is 20 and whose base is an equilateral triangle with sidelength 10?

Explanation:

The volume of a pyramid can be calculated using the formula

where  is the height and  is the area of the base.

Since the base is an equilateral triangle, its area can be calculated using the formula

Therefore, the volume can be rewritten as

Substitute :

### Example Question #7 : Calculating The Volume Of A Tetrahedron

A right pyramid has height ; its base is a square with four sides of length  each. What is the volume of this pyramid?

Explanation:

The volume of a pyramid can be calculated using the formula

where  is the area of the base and  is the height. The base of the pyramid in question is a square; if we let its sidelength be , this base will be , and the volume formula will be

Setting  and :

or

.