### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #65 : Solid Geometry

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

Give its volume.

**Possible Answers:**

**Correct answer:**

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 and height are both 18, so its area is

The fourth vertex is off the -plane; its perpendicular distance to the aforementioned face is its -coordinate, 9, so this is the height of the pyramid. The volume of the pyramid is

### Example Question #66 : Solid Geometry

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 .

**Possible Answers:**

**Correct answer:**

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

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

Its base is 12 and its height is 15, so its area is

The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, , so this is the height of the pyramid. The volume of the pyramid is

### Example Question #67 : Solid Geometry

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

Give its volume in terms of .

**Possible Answers:**

The correct answer is not among the other choices.

**Correct answer:**

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

Three of the vertices - - are on the horizontal plane , and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles (drawing not to scale):

Its base is 20 and its height is 9, so its area is

The fourth vertex is off this plane; its perpendicular (vertical) distance to the aforementioned face is the difference between the -coordinates, , so this is the height of the pyramid. The volume of the pyramid is

### Example Question #68 : Solid Geometry

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?

**Possible Answers:**

**Correct answer:**

The tetrahedron looks like this:

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

This is a triangular pyramid, so look at as its base; the area of the base is half the product of its legs, or

.

The volume of the tetrahedron, it being essentially a pyramid, is one third the product of its base and its height, the latter of which is 12. Therefore,

.

### Example Question #69 : Solid Geometry

Above is the base of a triangular pyramid, which is equilateral. , and the pyramid has height 30. What is the volume of the pyramid?

**Possible Answers:**

**Correct answer:**

Altitude divides into two 30-60-90 triangles.

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