GMAT Math : DSQ: Calculating the volume of a tetrahedron

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #171 : Geometry

 Tetra_4

The above figure shows a cube; a pyramid with vertices at  is inscribed. What is the volume of the pyramid?

Statement 1: The cube has volume 729.

Statement 2:  has area .

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Let  be the common length of the edges of the cube. 

The height of the pyramid is , and the area of its base, Square , is . The volume of the pyramid is one third the product of these, or 

Therefore, if the common length of the edges of the cube  can be found, the volume of the pyramid can be calculated.

From Statement 1 alone, this length can be calculated as the cube root of the volume of the cube, 729; this cube root is equal to 9.

From Statement 2 alone, this length can also be calculated.  has as one leg  with length ; its other leg, which is a diagonal of a square with sidelength  has length   by the 45-45-90 Theorem. The area is half the product of these legs; since this area is , we can find  as follows:

 

From either statement alone,  can be calculated to be 9, and the volume of the pyramid can be found to be 

Example Question #172 : Geometry

A regular tetrahedron has four equilateral faces and six congruent edges.

Of a given regular tetrahedron and a given right regular square pyramid, which, if either, has the greater volume?

Statement 1: The length of each side of the base of the square pyramid has the same length as one edge of the tetrahedron.

Statement 2: The height of the square pyramid has the same length of one edge of the tetrahedron.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Let  be the length of one edge of the regular tetrahedron. Its volume can be calculated using the formula

.

Now let  be the length of each side of the square base of the square pyramis, and  be the height of the square pyramid. Since the volume of this pyramid is one third the product of the height and the area of the base, the volume can be calculated using the formula

, or

Statement 1 alone tells us that , allowing the formula to be rewritten

,

but with no information about the height, we cannot compare this to tetrahedron height .

Likewise, 

Statement 1 alone tells us that , allowing the formula to be rewritten

,

but with no information about the sidelength or area of the base, we cannot compare this to the tetrahedron height.

Now assume both statements to be true. We know that , so the formula for the square pyramid is 

Since , it follows that:

and the square pyramid has the greater volume.

Example Question #181 : Geometry

A tetrahedron is a solid with four triangular faces.

Give the volume of a tetrahedron.

Statement 1: The tetrahedron has four equilateral faces.

Statement 2: The surface area of the tetrahedron is .

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Neither statement is enough to determine the volume of the tetrahedron; Statement 1 alone gives no actual measurements, and Statement 2 gives only the surface area, which can apply to infinitely many tetrahedrons.

Assume both statements to be true. A tetrahedron with four equilateral faces is a regular tetrahedron, whose surface area, relative to the common length  of its edges, is defined by the formula

.

By substituting  for , it is possible to calculate . Consequently, the volume of the tetrahedron can be calculated using the volume formula

.

Example Question #182 : Geometry

Tetra_3

Note: Figure NOT drawn to scale.

The above figure shows a rectangular prism with an inscribed tetrahedron, or triangular pyramid, with vertices . What is the volume of the tetrahedron?

Statement 1: Rectangle  has area 240.

Statement 2: Square  has area 144.

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The volume of the pyramid is one third the product of the height, which is , and the area of the base; this base, being a right triangle, is equal to one half the product of the lengths of its legs, or  and . Therefore, 

or 

From Statement 1, it can be determined that , but without knowing anything about , the volume of the pyramid cannot be determined. Similarly, from Statement 2, it can be determined that , but nothing is given about .

Now assume both statements are true. Statement 2 gives that Quadrilateral  is a square with area 144, so . From Statement 1, we can tell . The volume of the pyramid can be calculated as

.

Example Question #183 : Geometry

Two right regular pyramids, Pyramid 1 and Pyramid 2, have the same height. Do the pyramids have the same volume?

Statement 1: The bases of the two pyramids have the same area.

Statement 2: The base of Pyramid 1 has seven sides; the base of Pyramid 2 has nine sides.

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Correct answer:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Explanation:

The volume of any pyramid is one third the product of its height and the area of its base. Since the two pyramids have the same height, their volume will be the same if and only if their bases have the same area. Statement 1 gives us this information explicity. Statement 2 is irrelevant, since the number of sides of a polygon by itself does not give a clue as to its area.

Example Question #184 : Geometry

Tetra_3

Note: Figure NOT drawn to scale.

The above figure shows a rectangular prism with an inscribed tetrahedron, or triangular pyramid, with vertices . What is the volume of the tetrahedron?

Statement 1: 

Statement 2: 

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

We show that the two statements together provide insufficient information by assuming them both to be true.

The volume of the pyramid is one third the product of the height, which is , and the area of the base; this base, being a right triangle, is equal to one half the product of the lengths of its legs, or  and . Therefore, 

or 

By Statement 1, , and by Statement 2, , so by substitution,

Without any further information, however, the volume cannot be determined.

Example Question #185 : Geometry

A solid on the three-dimensional coordinate plane has four vertices, at points , and  for some positive values of  . What is the volume of the solid?

Statement 1: 

Statement 2: 

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

The figure described is the triangular pyramid, or tetrahedron, in the coordinate three-space below.

Tetrahedron

The base of the pyramid can be seen as a triangle with the three known coordinates ,  and , and the area of its base is half the product of the lengths of its legs, which is 

.

The volume of the pyramid is one third the product of the area of its base, which is 70, and its height, which is the perpendicular distance from the unknown point to the base. Since the base is entirely within the -plane, this distance is the -coordinate, which is . Therefore, the only thing that is needed to determine the volume of the pyramid is . However, the two statements together only yield , and therefore do not give sufficient information to solve the problem.

Example Question #186 : Geometry

Pyramid 1 and Pyramid 2 in three-dimensional coordinate space each have the same base: the square with vertices at the origin, , and . Pyramid 1 has its fifth vertex at the point ; Pyramid has its fifth vertex at the point . All six variables represent positive quantities. Which pyramid, if either, has the greater volume?

Statement 1: 

Statement 2: 

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The volume of a pyramid is one third the product of its height and the area its base. The two pyramids have the same base, so the pyramid with the greater height will have the greater volume (and if their heights are equal, their volumes are equal).

Pyramid 1 is shown below:

Pyramid

The base of the pyramid is on the -plane, so the height of the pyramid  is the perpendicular distance from apex  to this plane; this is the -coordinate, . Pyramid 2 has the same base and apex , so its height is the -coordinate of its apex, 

Therefore, whichever is greater -  or  - determines which pyramid has the greater volume.

Neither statement alone gives a clue as to which is greater. However, if we assume both, then, by the subtraction property of inequality,

and

together imply that

and

.

This means that Pyramid 1 has the greater height and, consequently, the greater volume.

Example Question #11 : Rectangular Solids & Cylinders

Do right regular pyramids Pyramid 1 and Pyramid 2 have the same volume?

Statement 1: The bases of the two pyramids have the same perimeter.

Statement 2: The two pyramids have the same height.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question. 

Explanation:

The volume of any pyramid is one third the product of its height and the area of its base. Also, the base of a right regular pyramid must be a regular polygon.

Assume both statements are true. Since, from Statement 2, both pyramids have the same height, their volumes are equal if and only if the areas of their bases are equal as well.

However, from Statement 1, the perimeters of the bases are what are given as being equal. Two regular poygons with the same perimeter and the same number of sides have the same area. But as exemplified below, two regular polygons with different numbers of sides and the same perimeter have different areas. 

For example, an equilateral triangle with perimeter 60 has sidelength 20 and, consequently, area

A square with perimeter 60 has sidelength 15, and, consequently, area equal to the square of this, 225.

Example Question #188 : Geometry

A solid on the three-dimensional coordinate plane has four vertices, at points , and  for some positive values of  . What is the volume of the solid?

Statement 1: 

Statement 2: 

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question. 

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

The figure described is the triangular pyramid, or tetrahedron, in the coordinate three-space below.

Tetrahedron

The base of the pyramid can be seen as a triangle with the three known coordinates ,  and , and the area of its base is half the product of the lengths of its legs, which is 

.

The volume of the pyramid is one third the product of the area of its base, which is 70, and its height, which is the perpendicular distance from the unknown point to the base. Since the base is entirely within the -plane, this distance is the -coordinate, which is . Therefore, the only thing that is needed to determine the volume of the pyramid is .

Neither statement alone is enough to gain this information. However, if both statements are assumed true, we can subtract each side of the latter equation from the former as follows:

              

                 

The value of  is obtained and the volume of the pyramid can be calculated.

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