GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2401 : Gmat Quantitative Reasoning

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 $120 \sqrt{3}$ .

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.

EITHER 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.

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:

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

$A = s^{2}\sqrt{3}$ .

By substituting $120 \sqrt{3}$ for , it is possible to calculate . Consequently, the volume of the tetrahedron can be calculated using the volume formula

$V = \frac{s^{3} }{6\sqrt{2}}$ .

Report an Error

Example Question #294 : Data Sufficiency Questions

Tetra_3

Note: Figure NOT drawn to scale.

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

Statement 1: Rectangle ADHE has area 240.

Statement 2: Square ABFE has area 144.

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 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,

$V = \frac{1}{3} \cdot AE \cdot \frac{1}{2} EF \cdot EH$

$V = \frac{1}{6} \cdot AE \cdot EH \cdot EF$

From Statement 1, it can be determined that $AE \cdot EH = 240$ , but without knowing anything about , the volume of the pyramid cannot be determined. Similarly, from Statement 2, it can be determined that $AE \cdot EF = 144$ , but nothing is given about .

Now assume both statements are true. Statement 2 gives that Quadrilateral ABFE is a square with area 144, so $AE = EF = \sqrt{144} = 12$ . From Statement 1, we can tell $AE \cdot EH = 240$ . The volume of the pyramid can be calculated as

$V = \frac{1}{6} \cdot AE \cdot EH \cdot EF = \frac{1}{6}\cdot240 \cdot 12= 480$ .

Report an Error

Example Question #181 : 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:

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.

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.

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.

Report an Error

Example Question #12 : Dsq: Calculating The Volume Of A Tetrahedron

Tetra_3

Note: Figure NOT drawn to scale.

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

Statement 1: AE = EF

Statement 2: EH = 40

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 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 insufficient to answer the question.

Explanation:

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

$V = \frac{1}{3} \cdot AE \cdot \frac{1}{2} EF \cdot EH$

$V = \frac{1}{6} \cdot AE \cdot EF \cdot EH$

By Statement 1, AE=EF , and by Statement 2, EH = 40 , so by substitution,

$V = \frac{1}{6} \cdot AE \cdot AE \cdot 40$

$V = \frac{20}{3} \cdot (AE )^{2}$

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

Report an Error

Example Question #297 : Data Sufficiency Questions

A solid on the three-dimensional coordinate plane has four vertices, at points (0,0,0) , (10,0,0) , (0,14,0) , and (A,B,C) for some positive values of A,B,C . What is the volume of the solid?

Statement 1: A=C

Statement 2: B = 8

Possible Answers:

EITHER 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.

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.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is 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 (0,0,0) , (10,0,0) , and (0,14,0) , and the area of its base is half the product of the lengths of its legs, which is

$b = \frac{1}{2} \cdot 10 \cdot 14 = 70$ .

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.

Report an Error

Example Question #301 : Data Sufficiency Questions

Pyramid 1 and Pyramid 2 in three-dimensional coordinate space each have the same base: the square with vertices at the origin, (10,0,0) , (10,10,0) , and (0,10,0) . Pyramid 1 has its fifth vertex at the point (D,E,F) ; Pyramid has its fifth vertex at the point (G,H,J) . All six variables represent positive quantities. Which pyramid, if either, has the greater volume?

Statement 1: D+E+F > G+H+J

Statement 2: D+E= G+H

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

EITHER 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.

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.

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 (D,E,F) to this plane; this is the -coordinate, . Pyramid 2 has the same base and apex (G,H,J) , 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,

D+E+F > G+H+J

and

D+E= G+H

together imply that

$D+E+F - ( D+E) > G+H+J - \left ( G+H \right )$

and

F > J .

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

Report an Error

Example Question #12 : Tetrahedrons

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:

EITHER 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.

BOTH statements TOGETHER are insufficient 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.

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

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{20^{2}\sqrt{3}}{4} = 100\sqrt{3}$

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

Report an Error

Example Question #20 : Dsq: Calculating The Volume Of A Tetrahedron

A solid on the three-dimensional coordinate plane has four vertices, at points (0,0,0) , (10,0,0) , (0,14,0) , and (A,B,C) for some positive values of A,B,C . What is the volume of the solid?

Statement 1: A+B+C= 50

Statement 2: A+B-C= 30

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

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 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 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

$b = \frac{1}{2} \cdot 10 \cdot 14 = 70$ .

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:

A+B+C= 50

$\underline{A+B-C= 30}$

2C = 20

C= 10

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

Report an Error

Example Question #21 : Tetrahedrons

Pyramid 1 in three-dimensional coordinate space has as its base the square with vertices at the origin, (10,0,0) , (10,10,0) , and (0,10,0) , and its apex at the point (D,E,F) ; Pyramid 2 has as its base the square with vertices at the origin, (12,0,0) , (12,12,0) , and (0,12,0) , and its apex at the point (G,H,J) . All six variables represent positive quantities. Which pyramid has the greater volume?

Statement 1: $10 \le F \le 12$ and $10 \le J \le 12$

Statement 2: F > J

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.

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 volume of a pyramid is one third the product of its height and the area its base.

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 (D,E,F) to this plane; this is the -coordinate, . The base of the pyramid is a square of sidelength 10, so its area is the square of 10, or 100. This makes the volume of Pyramid 1

$V_{1} = \frac{1}{3} \cdot 100 \cdot F = \frac{100}{3}F = \left ( 33 \frac{1}{3}\right )F$

Similarly, the volume of Pyramid 2 is

$V_{2} = \frac{1}{3} \cdot 12 ^{2}\cdot J= \frac{1}{3} \cdot 144 \cdot J = 36 J$

Therefore, the problem asks us to determine which of $\left ( 33 \frac{1}{3}\right )F$ and 36J is the greater.

We show that the two statements together provide insufficient information by examining two scenarios.

Case 1: F =12, J = 10

$V_{1} = \left ( 33 \frac{1}{3}\right )F = \left ( 33 \frac{1}{3}\right ) \cdot 12 = 400$

$V_{2} = 36 J = 36 \cdot 10 = 360$

$V_{1} > V_{2}$

Case 2: $F =10 \frac{1}{6}, J = 10$

$V_{1} = \left ( 33 \frac{1}{3}\right )F = \left ( 33 \frac{1}{3}\right ) \cdot 10 \frac{1}{6} = 338 \frac{8}{9}$

$V_{2} = 36 J = 36 \cdot 10 = 360$

$V_{1}< V_{2}$

Each case fits the conditions of the two statements and the main body of the question; in one case, Pyramid 1 has the greater volume and in the other case, Pyramid 2 does.

Report an Error

Example Question #22 : Tetrahedrons

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 perimeter.

Statement 2: The base of Pyramid 1 is a hexagon; the base of Pyramid 2 is an octagon.

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

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.

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 volume of any pyramid is one third the product of its height and the area of its base. Since both pyramids have the same height, their volumes are equal if and only if the areas of their bases are equal as well. Also, since both pyramids are regular, their bases are regular polygons.

Statement 1 alone is insufficient. It is possible for two regular polygons to have the same perimeter but different areas. For example, an equilateral triangle with perimeter 60 has sidelength 20 and, consequently, area

$A = \frac{s^{2}\sqrt{3}}{4} = \frac{20^{2}\sqrt{3}}{4} = 100\sqrt{3}$

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

Statement 2 only gives the number of sides, and no information about their measures.

Now assume both statements are true. If the common perimeter is , then the length of one side of the base of Pyramid 1, it being a regular hexagon, is $\frac{1}{6}P$ , likewise, the length of one side of the base of Pyramid 2 is $\frac{1}{8}P$ . The area of each in terms of can be calculated, and the two can be compared.

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

SSAT Tutors in Los Angeles, GMAT Tutors in Los Angeles, Chemistry Tutors in Philadelphia, English Tutors in Houston, SAT Tutors in Los Angeles, GRE Tutors in Miami, Computer Science Tutors in Atlanta, Math Tutors in Dallas Fort Worth, ACT Tutors in Philadelphia, SSAT Tutors in Seattle

Popular Courses & Classes

SSAT Courses & Classes in Denver, ACT Courses & Classes in Denver, SAT Courses & Classes in Boston, ACT Courses & Classes in Chicago, LSAT Courses & Classes in San Diego, SAT Courses & Classes in Miami, ACT Courses & Classes in Philadelphia, ISEE Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in New York City, SSAT Courses & Classes in San Diego

Popular Test Prep

ACT Test Prep in Denver, ACT Test Prep in Boston, GRE Test Prep in Dallas Fort Worth, ACT Test Prep in Chicago, ISEE Test Prep in Miami, MCAT Test Prep in San Diego, LSAT Test Prep in Denver, LSAT Test Prep in San Francisco-Bay Area, ACT Test Prep in New York City, SAT Test Prep in Los Angeles

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

Email address:
Your name:
Feedback:

GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #2401 : Gmat Quantitative Reasoning

Example Question #294 : Data Sufficiency Questions

Example Question #181 : Geometry

Example Question #12 : Dsq: Calculating The Volume Of A Tetrahedron

Example Question #297 : Data Sufficiency Questions

Example Question #301 : Data Sufficiency Questions

Example Question #12 : Tetrahedrons

Example Question #20 : Dsq: Calculating The Volume Of A Tetrahedron

Example Question #21 : Tetrahedrons

Example Question #22 : Tetrahedrons

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details