ISEE Upper Level Math : Tetrahedrons

Study concepts, example questions & explanations for ISEE Upper Level Math

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Example Questions

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Example Question #1 : Tetrahedrons

A triangular pyramid, or tetrahedron, with volume 100 has four vertices with Cartesian coordinates 

where .

Evaluate

Possible Answers:

Correct answer:

Explanation:

The tetrahedron is as follows (figure not to scale):

Tetrahedron

This is a triangular pyramid with a right triangle with legs 10 and  as its base; the area of the base is 

 

The height of the pyramid is 5, so

Set this equal to 100 to get :

 

Example Question #2 : Tetrahedrons

A triangular pyramid, or tetrahedron, with volume 1,000 has four vertices with Cartesian coordinates 

where .

Evaluate 

Possible Answers:

Correct answer:

Explanation:

The tetrahedron is as follows:

Pyramid

This is a triangular pyramid with a right triangle with two legs of measure  as its base; the area of the base is 

 

Since the height of the pyramid is also , the volume is

.

Set this equal to 1,000:

Example Question #1 : How To Find The Length Of An Edge Of A Tetrahedron

A triangular pyramid, or tetrahedron, with volume 240 has four vertices with Cartesian coordinates 

where .

Evaluate 

Possible Answers:

Correct answer:

Explanation:

The tetrahedron is as follows (figure not to scale):

Tetrahedron

This is a triangular pyramid with a right triangle with two legs of measure  as its base; the area of the base is 

 

The height of the pyramid is 24, so the volume is

Set this equal to 240 to get :

Example Question #4 : Tetrahedrons

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:

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:

Thingy

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 #381 : 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:

Explanation:

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:

Thingy

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 #61 : 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:

Explanation:

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):

Thingy

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 #7 : Tetrahedrons

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:

Explanation:

The tetrahedron looks like this:

Tetrahedron

 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 #8 : Tetrahedrons

Thingy

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:

Explanation:

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:

Example Question #62 : Solid Geometry

A regular tetrahedron has edges of length 4. What is its surface area?

Possible Answers:

Correct answer:

Explanation:

A regular tetrahedron has four faces, each of which is an equilateral triangle. Therefore, its surface area, given sidelength , is 

.

Substitute :

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

A regular tetrahedron comprises four faces, each of which is an equilateral triangle. Each edge has length 16. What is its surface area?

Possible Answers:

Correct answer:

Explanation:

The area of each face of a regular tetrahedron, that face being an equilateral triangle, is 

Substitute edge length 16 for :

The tetrahedron has four faces, so the total surface area is 

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