GMAT Math : Calculating the volume of a tetrahedron

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 #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.

Possible Answers:

125

$125\sqrt{3}$

375

250

$375\sqrt{3}$

Correct answer:

$125\sqrt{3}$

Explanation:

The base of the triangle has an area that can be found using the formula for the area of an equilateral triangle, substituting s=10 :

$B = \frac{s^{2}\sqrt{3}}{4}$

$B = \frac{10^{2}\sqrt{3}}{4}$

$B= \frac{100\sqrt{3}}{4}$

$B= 25\sqrt{3}$

Now, in the formula for the volume of a pyramid, substitute $B = 25\sqrt{3}, h = 15$ :

$V = \frac{1}{3} Bh$

$V = \frac{1}{3}\cdot 25\sqrt{3} \cdot 15$

$V = 125\sqrt{3}$

Report an Error

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.

Possible Answers:

$243 \textrm{ in}^{3}$

$72 \textrm{ in}^{3}$

$216 \textrm{ in}^{3}$

$729 \textrm{ in}^{3}$

$108 \textrm{ in}^{3}$

Correct answer:

$243 \textrm{ in}^{3}$

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 $9 ^{2} = 81$ square inches. The volume of a pyramid is one-third the product of its height and the area of its base, so substitute B = 81, h = 9 in the following:

$V = \frac{1}{3} Bh = \frac{1}{3} \cdot 81 \cdot 9 = 243$ cubic inches.

Report an Error

Example Question #1 : Calculating The Volume Of A Tetrahedron

In three-dimensional space, the four vertices of a tetrahedron - a solid with four faces - have Cartesian coordinates (0,0,0), (60,0,0), (30, 40, 0), (40,30,20) .

Give its volume.

Possible Answers:

8,000

16,000

24,000

32,000

12,000

Correct answer:

8,000

Explanation:

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

Three of the vertices - (0,0,0), (60,0,0), (30, 40, 0) - are on the -plane, and can be seen as the vertices of the triangular base. This triangle, as seen below, is isosceles:

Base

Its base is 60 and its height is 40, so its area is

$B = \frac{1}{2} \cdot 60 \cdot 40 = 1,200$

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

$V = \frac{1}{3} \cdot 1,200 \cdot 20 = 8,000$ .

Report an Error

Example Question #483 : Geometry

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

(0,0,0), (n ,0,0), (0,4n,0), (0,0, 9n) ,

where n > 1

Give its volume in terms of .

Possible Answers:

$6 n^{3}$

$9n^{3}$

$\small 4 n^{3}$

$n^{3}$

$3 n^{3}$

Correct answer:

$6 n^{3}$

Explanation:

The tetrahedron looks like this:

Tetrahedron

is the origin and A,B,C 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 $\Delta OCB$ the (right triangular) base; its area is half the product of its legs, or

$B = \frac{1}{2} \cdot n \cdot 4n= 2n^{2}$ .

The volume of the tetrahedron is one third the product of its base and its height, the latter of which is . Therefore,

$V = \frac{1}{3} \cdot 9n \cdot 2n^{2} = 6 n^{3}$ .

Report an Error

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 (0,0,0), (60,0,0), (0,60,0), (0,0,60) .

What is the volume of this tetrahedron?

Possible Answers:

108,000

36,000

54,000

18,000

72,000

Correct answer:

36,000

Explanation:

The tetrahedron looks like this:

Tetrahedron

is the origin and A,B,C 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 $\Delta OAB$ the (right triangular) base; its area is half the product of its legs, or

$B = \frac{1}{2} \cdot 60 \cdot 60 = 1,800$ .

The volume of the tetrahedron is one third the product of its base and its height, the latter of which is 60. Therefore,

$V = \frac{1}{3} \cdot 60 \cdot 1,800 = 36,000$ .

Report an Error

Example Question #3 : 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?

Possible Answers:

1,000

$\frac{500 \sqrt{3}}{3}$

$500 \sqrt{3}$

2,000

$\frac{1.000 \sqrt{3}}{3}$

Correct answer:

$\frac{500 \sqrt{3}}{3}$

Explanation:

The volume of a pyramid can be calculated using the formula

$V = \frac{1}{3}Bh$

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

$B = \frac{s^{2} \sqrt{3}}{4}$

Therefore, the volume can be rewritten as

$V = \frac{1}{3} \cdot \frac{s^{2} \sqrt{3}}{4} \cdot h$

Substitute s = 10 , h = 20 :

$V = \frac{1}{3} \cdot \frac{10^{2}\cdot \sqrt{3}}{4} \cdot 20$

$= \frac{1}{3} \cdot \frac{100\cdot \sqrt{3}}{4} \cdot\frac{ 20}{1}$

$= \frac{1}{3} \cdot \frac{25\cdot \sqrt{3}}{1} \cdot\frac{ 20}{1}$

$=\frac{500 \sqrt{3}}{3}$

Report an Error

Example Question #2 : Calculating The Volume Of A Tetrahedron

A right pyramid has height $3^{x}$ ; its base is a square with four sides of length $3^{y}$ each. What is the volume of this pyramid?

Possible Answers:

$3 ^{2x+y-1 }$

$3 ^{x+2y-1 }$

$3 ^{x+2y+1 }$

$3 ^{2x+y+1 }$

$3 ^{x+4y-1 }$

Correct answer:

$3 ^{x+2y-1 }$

Explanation:

The volume of a pyramid can be calculated using the formula

$V = \frac{1}{3} Bh$ ,

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 $s^{2}$ , and the volume formula will be

$V = \frac{1}{3} s ^{2}h$

Setting $s = 3^{y}$ and $h = 3^{x}$ :

$V =3 ^{-1} \cdot ( 3^{y}) ^{2} \cdot 3^{x}$

$V =3 ^{-1} \cdot 3^{2 y} \cdot 3^{x}$

$V =3 ^{-1+2 y+x}$

$V =3 ^{x+2y-1 }$ .

Report an Error

All GMAT Math Resources

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

Popular Subjects

MCAT Tutors in Phoenix, Algebra Tutors in Atlanta, MCAT Tutors in San Diego, Spanish Tutors in Denver, Algebra Tutors in San Francisco-Bay Area, Spanish Tutors in Phoenix, Chemistry Tutors in Los Angeles, GMAT Tutors in San Francisco-Bay Area, Math Tutors in Miami, French Tutors in Houston

Popular Courses & Classes

ISEE Courses & Classes in Los Angeles, MCAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Phoenix, GRE Courses & Classes in Houston, MCAT Courses & Classes in Houston, GMAT Courses & Classes in Chicago, MCAT Courses & Classes in Atlanta, ACT Courses & Classes in Los Angeles, SSAT Courses & Classes in San Diego, SAT Courses & Classes in Philadelphia

Popular Test Prep

SSAT Test Prep in Dallas Fort Worth, GMAT Test Prep in Denver, SAT Test Prep in San Francisco-Bay Area, SSAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Denver, LSAT Test Prep in Houston, GMAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Washington DC, ACT Test Prep in Atlanta, GRE Test Prep in Chicago

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 : Calculating the volume of a tetrahedron

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Calculating The Volume Of A Tetrahedron

Example Question #2 : Calculating The Volume Of A Tetrahedron

Example Question #1 : Calculating The Volume Of A Tetrahedron

Example Question #483 : Geometry

Example Question #3 : Calculating The Volume Of A Tetrahedron

Example Question #3 : Calculating The Volume Of A Tetrahedron

Example Question #2 : Calculating The Volume Of A Tetrahedron

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details