It is currently 23 Nov 2017, 12:04

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Math : 3-D Geometries

Author Message
TAGS:

### Hide Tags

Retired Moderator
Joined: 02 Sep 2010
Posts: 793

Kudos [?]: 1212 [27], given: 25

Location: London

### Show Tags

01 Oct 2010, 18:18
27
KUDOS
34
This post was
BOOKMARKED
3-D Geometries

This post is a part of [GMAT MATH BOOK]

created by: shrouded1

Scope

The GMAT often tests on the knowledge of the geometries of 3-D objects such cylinders, cones, cubes & spheres. The purpose of this document is to summarize some of the important ideas and formulae and act as a useful cheat sheet for such questions

Cube
Attachment:

cube.jpg [ 8.97 KiB | Viewed 4573 times ]

A cube is the 3-D generalisation of a square, and is characterized by the length of the side, $$a$$. Important results include :

• Volume = $$a^3$$
• Surface Area = $$6a^2$$
• Diagnol Length = $$\sqrt{3}a$$

Cuboid

Attachment:

cuboid.jpg [ 16.29 KiB | Viewed 4577 times ]

A cube is the 3-D generalisation of a rectangle, and is characterized by the length of its sides, $$a,b,c$$. Important results include :

• Volume = $$abc$$
• Surface Area = $$2(ab+bc+ca)$$
• Diagnol Length = $$\sqrt{a^2+b^2+c^2}$$

Cylinder

Attachment:

cylinder.jpg [ 24.95 KiB | Viewed 4572 times ]

A cylinder is a 3-D object formed by rotating a rectangular sheet along one of its sides. It is characterized by the radius of the base, $$r$$, and the height, $$h$$. Important results include :

• Volume = $$\pi r^2 h$$
• Outer surface area w/o bases = $$2 \pi r h$$
• Outer surface area including bases = $$2 \pi r (r+h)$$

Cone

Attachment:

cone.jpg [ 31.94 KiB | Viewed 4575 times ]

A cone is a 3-D object obtained by rotating a right angled triangle around one of its sides. It is charcterized by the radius of its base, $$r$$, and the height, $$h$$. The hypotenuse of the triangle formed by the height and the radius (running along the diagnol side of the cone), is known as it lateral height, $$l=\sqrt{r^2+h^2}$$. Important results include :

• Volume = $$\frac{1}{3} \pi r^2 h$$
• Outer surface area w/o base = $$\pi r l =\pi r \sqrt{r^2+h^2}$$
• Outer surface area including base = $$\pi r (r+l)=\pi r (r+\sqrt{r^2+h^2})$$

Sphere

Attachment:

sphere.jpg [ 32.94 KiB | Viewed 4572 times ]

A sphere is a 3-D generalisation of a circle. It is characterised by its radius, $$r$$. Important results include :

• Volume = $$\frac{4}{3} \pi r^3$$
• Surface Area= $$4 \pi r^2$$

Attachment:

hemisphere.jpg [ 36.64 KiB | Viewed 4567 times ]

A hemisphere is a sphere cut in half and is also characterised by its radius $$r$$. Important results include :

• Volume = $$\frac{2}{3} \pi r^3$$
• Surface Area w/o base = $$2 \pi r^2$$
• Surface Area with base = $$3 \pi r^2$$

Some simple configurations

These may appear in various forms on the GMAT, and are good practice to derive on one's own :

1. Sphere inscribed in cube of side $$a$$ : Radius of sphere is $$\frac{a}{2}$$
2. Cube inscribed in sphere of radius $$r$$ : Side of cube is $$\frac{2r}{\sqrt{3}}$$
3. Cylinder inscribed in cube of side $$a$$ : Radius of cylinder is $$\frac{a}{2}$$; Height $$a$$
4. Cone inscribed in cube of side $$a$$ : Radius of cone is $$\frac{a}{2}$$; Height $$a$$
5. Cylinder of radius $$r$$ in sphere of radius $$R$$ ($$R>r$$) : Height of cylinder is $$2\sqrt{R^2-r^2}$$

Examples

Example 1 : A certain right circular cylinder has a radius of 5 inches. There is oil filled in this cylinder to the height of 9 inches. If the oil is poured completely into a second right cylinder, then it will fill the second cylinder to a height of 4 inches. What is the radius of the second cylinder, in inches?

A. 6
B. 6.5
C. 7
D. 7.5
E. 8

Solution : The volume of the liquid is constant.
Initial volume = $$\pi * 5^2 * 9$$
New volume = $$\pi * r^2 * 4$$
$$\pi * 5^2 * 9 = \pi * r^2 * 4$$
$$r = (5*3)/2 = 7.5$$

Example 2 : A spherical balloon has a volume of 972 $$\pi$$cubic cm, what is the surface area of the balloon in sq cm?

A) 324
B) 729
C) 243 $$\pi$$
D) 324 $$\pi$$
E) 729 $$\pi$$

Solution : $$V=\frac{4}{3} \pi r^3$$
$$r = (\frac{3V}{4\pi})^{\frac{1}{3}} = (\frac{3 * 972 * \pi}{4 * \pi})^{\frac{1}{3}} = (3*243)^{1/3} = (3^6)^{1/3} = 9$$
$$A=4 \pi r^2 = 4 * \pi * 9^2=324 \pi$$

Example 3 : A cube of side 5cm is painted on all its side. If it is sliced into 1 cubic centimer cubes, how many 1 cubic centimeter cubes will have exactly one of their sides painted?

A. 9
B. 61
C. 98
D. 54
E. 64

Solution : Notice that the new cubes will be each of side 1Cm. So on any face of the old cube there will be 5x5=25 of the smaller cubes. Of these, any smaller cube on the edge of the face will have 2 faces painted (one for every face shared with the bigger cube). The number of cubes that have exacly one face painted are all except the ones on the edges. Number on the edges are 16, so 9 per face.

There are 6 faces, hence 6*9=54 smaller cubes with just one face painted.

Example 4 : What is the surface area of the cuboid C ?
(1) The length of the diagnol of C is 5
(2) The sum of the sides of C is 10

Solution : Let the sides of cuboid C be $$x,y,z$$
We know that the surface area is given be $$2(xy+yz+zx)$$
(1) : Diagnol = $$\sqrt{x^2+y^2+z^2}=5$$. Not sufficient to know the area
(2) : Sum of sides = $$x+y+z=10$$. Not sufficient to know the area
(1+2) : Note the identity $$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)$$
Now we clearly have enough information.
$$2(xy+yz+zx) = 10^2 - 5^2 = 75$$
Sufficient

3-D Geometry Questions:

http://gmatclub.com/forum/3-d-geometry- ... 71024.html

Some Other 3-D Problems

Sphere & Cube
Sphere & Cylinder
Cylinder & Cuboid
Cylinder & Cuboid II
Cylinder
Cube
Cube II
Cone
Cube III
Cylinder
Hemisphere
_________________

Last edited by bb on 29 Jan 2017, 00:13, edited 8 times in total.

Kudos [?]: 1212 [27], given: 25

Intern
Joined: 11 Aug 2010
Posts: 23

Kudos [?]: 33 [2], given: 37

Schools: SUNY at Stonyb Brook
WE 1: 4 yrs
Re: Math : 3-D Geometries [#permalink]

### Show Tags

01 Oct 2010, 19:51
2
KUDOS
Thanks a ton for posting this +1 Kudo
_________________

Consider giving Kudos if my post helped you in some way

Kudos [?]: 33 [2], given: 37

Retired Moderator
Joined: 02 Sep 2010
Posts: 793

Kudos [?]: 1212 [3], given: 25

Location: London
Re: Math : 3-D Geometries [#permalink]

### Show Tags

02 Oct 2010, 09:16
3
KUDOS
Added some solved examples to the post
_________________

Kudos [?]: 1212 [3], given: 25

Retired Moderator
Status: I wish!
Joined: 21 May 2010
Posts: 784

Kudos [?]: 484 [0], given: 33

Re: Math : 3-D Geometries [#permalink]

### Show Tags

02 Oct 2010, 10:32
Wow.. amazing stuff. +1
_________________

http://drambedkarbooks.com/

Kudos [?]: 484 [0], given: 33

Founder
Joined: 04 Dec 2002
Posts: 15883

Kudos [?]: 29115 [1], given: 5272

Location: United States (WA)
GMAT 1: 750 Q49 V42
Re: Math : 3-D Geometries [#permalink]

### Show Tags

02 Oct 2010, 23:39
1
KUDOS
Expert's post
WOOOOOOOOOOOOOOOOOOOOOOOOOOOOOWWWWWWWWWWWWWWWWWWWW

I sense Bunuel has competition!
_________________

Founder of GMAT Club

Just starting out with GMAT? Start here... or use our Daily Study Plan

Co-author of the GMAT Club tests

Kudos [?]: 29115 [1], given: 5272

Current Student
Status: Three Down.
Joined: 09 Jun 2010
Posts: 1914

Kudos [?]: 2230 [0], given: 210

Concentration: General Management, Nonprofit
Re: Math : 3-D Geometries [#permalink]

### Show Tags

03 Oct 2010, 00:09
Mathematica be the shizzles.

Kudos [?]: 2230 [0], given: 210

Senior Manager
Joined: 21 Sep 2010
Posts: 261

Kudos [?]: 35 [0], given: 56

Re: Math : 3-D Geometries [#permalink]

### Show Tags

08 Oct 2010, 05:38
Thank you very much!
_________________

"Only by going too far, can one find out how far one can go."

--T.S. Elliot

Kudos [?]: 35 [0], given: 56

Senior Manager
Status: Not afraid of failures, disappointments, and falls.
Joined: 20 Jan 2010
Posts: 290

Kudos [?]: 266 [0], given: 260

Concentration: Technology, Entrepreneurship
WE: Operations (Telecommunications)
Re: Math : 3-D Geometries [#permalink]

### Show Tags

09 Oct 2010, 07:01
Kudos! for writing that up. 3-D is what was missing in GMAT Math Book, gotta update it in compiled GMAT Math Book pdf.
_________________

"I choose to rise after every fall"
Target=770
http://challengemba.blogspot.com
Kudos??

Kudos [?]: 266 [0], given: 260

Intern
Joined: 24 May 2011
Posts: 4

Kudos [?]: [0], given: 0

Re: Math : 3-D Geometries [#permalink]

### Show Tags

03 Jun 2011, 10:17
Thnks for sharing....

Kudos [?]: [0], given: 0

Manager
Joined: 23 Aug 2011
Posts: 78

Kudos [?]: 277 [0], given: 13

Re: Math : 3-D Geometries [#permalink]

### Show Tags

25 Aug 2012, 01:34
Thanks a tons for this wonderful post.

This question/ configuration is quite popular(already explained by you in other threads), you might include it as well.
Inscribed Sphere touching the edges of cube.
Cube edge length=a; Radius of sphere=a/2; diagnoal of cube =(3^1/2)*a
The shortest length from edge of the cube to sphere's surface is given by
half the diagnol of cubeminus the raduis of sphere.

[(3^1/2)*a]/2-a/2 =a/2*[(3^1/2)-1]
Attachments

sphere.png [ 19.43 KiB | Viewed 26573 times ]

_________________

Whatever one does in life is a repetition of what one has done several times in one's life!
If my post was worth it, then i deserve kudos

Kudos [?]: 277 [0], given: 13

Math Expert
Joined: 02 Sep 2009
Posts: 42339

Kudos [?]: 133136 [0], given: 12415

Re: Math : 3-D Geometries [#permalink]

### Show Tags

11 Jul 2013, 00:07
Bumping for review*.

*New project from GMAT Club!!! Check HERE

_________________

Kudos [?]: 133136 [0], given: 12415

Intern
Joined: 20 May 2014
Posts: 35

Kudos [?]: 8 [0], given: 1

Re: Math : 3-D Geometries [#permalink]

### Show Tags

01 Jul 2014, 09:13
m]r = (\frac{3V}{4\pi})^{\frac{1}{3}} = (\frac{3 * 972 * \pi}{4 * \pi})^{\frac{1}{3}} = (3*243)^{1/3} = (3^6)^{1/3} = 9[/m]

How exactly did you get from (\frac{3 * 972 * \pi}{4 * \pi})^{\frac{1}{3}} to this ---> = (3*243)^{1/3}

Do we have to multiply the 3 and 972 inside the parntheses? and what happens to the 4?

Thanks

Kudos [?]: 8 [0], given: 1

Math Expert
Joined: 02 Sep 2009
Posts: 42339

Kudos [?]: 133136 [0], given: 12415

Re: Math : 3-D Geometries [#permalink]

### Show Tags

01 Jul 2014, 09:15
sagnik2422 wrote:
m]r = (\frac{3V}{4\pi})^{\frac{1}{3}} = (\frac{3 * 972 * \pi}{4 * \pi})^{\frac{1}{3}} = (3*243)^{1/3} = (3^6)^{1/3} = 9[/m]

How exactly did you get from (\frac{3 * 972 * \pi}{4 * \pi})^{\frac{1}{3}} to this ---> = (3*243)^{1/3}

Do we have to multiply the 3 and 972 inside the parntheses? and what happens to the 4?

Thanks

_________________

Kudos [?]: 133136 [0], given: 12415

Intern
Joined: 20 May 2014
Posts: 35

Kudos [?]: 8 [0], given: 1

Re: Math : 3-D Geometries [#permalink]

### Show Tags

01 Jul 2014, 09:27
A cube of side 5cm is painted on all its side. If it is sliced into 1 cubic centimer cubes, how many 1 cubic centimeter cubes will have exactly one of their sides painted?

A. 9
B. 61
C. 98
D. 54
E. 64

Solution : Notice that the new cubes will be each of side 1Cm. So on any face of the old cube there will be 5x5=25 of the smaller cubes. Of these, any smaller cube on the edge of the face will have 2 faces painted (one for every face shared with the bigger cube). The number of cubes that have exacly one face painted are all except the ones on the edges. Number on the edges are 16, so 9 per face.

There are 6 faces, hence 6*9=54 smaller cubes with just one face painted.

QUESTION : Number on the edges are 16, so 9 per face.

HOW DO WE KNOW THERE ARE 16 EDGES ? AND FROM THIS HOW IS 9 CALCULATED ?

THANKS

Kudos [?]: 8 [0], given: 1

Math Expert
Joined: 02 Sep 2009
Posts: 42339

Kudos [?]: 133136 [0], given: 12415

Re: Math : 3-D Geometries [#permalink]

### Show Tags

01 Jul 2014, 10:24
sagnik2422 wrote:
A cube of side 5cm is painted on all its side. If it is sliced into 1 cubic centimer cubes, how many 1 cubic centimeter cubes will have exactly one of their sides painted?

A. 9
B. 61
C. 98
D. 54
E. 64

Solution : Notice that the new cubes will be each of side 1Cm. So on any face of the old cube there will be 5x5=25 of the smaller cubes. Of these, any smaller cube on the edge of the face will have 2 faces painted (one for every face shared with the bigger cube). The number of cubes that have exacly one face painted are all except the ones on the edges. Number on the edges are 16, so 9 per face.

There are 6 faces, hence 6*9=54 smaller cubes with just one face painted.

QUESTION : Number on the edges are 16, so 9 per face.

HOW DO WE KNOW THERE ARE 16 EDGES ? AND FROM THIS HOW IS 9 CALCULATED ?

THANKS

This solution should be edited.

A cube has 12 edges, 6 faces and 8 vertices:
Attachment:

faces-edges-vertices.png [ 13.61 KiB | Viewed 18933 times ]

As for the question. Look at the image below:
Attachment:

MagicCube5x5.jpg [ 71.65 KiB | Viewed 23649 times ]
Little cubes with exactly one painted side will be those 3*3=9, which are in the center of each face. (6 faces)*(9 per each) = 54.

Similar questions to practice:
the-entire-exterior-of-a-large-wooden-cube-is-painted-red-155955.html
a-big-cube-is-formed-by-rearranging-the-160-coloured-and-99424.html
64-small-identical-cubes-are-used-to-form-a-large-cube-151009.html
a-wooden-cube-whose-edge-length-is-10-inches-is-composed-of-162570.html
if-a-4-cm-cube-is-cut-into-1-cm-cubes-then-what-is-the-107843.html
a-large-cube-consists-of-125-identical-small-cubes-how-110256.html

3-D Geometry Questions to practice: 3-d-geometry-questions-171024.html
_________________

Kudos [?]: 133136 [0], given: 12415

Intern
Joined: 19 Feb 2015
Posts: 1

Kudos [?]: [0], given: 1

Re: Math : 3-D Geometries [#permalink]

### Show Tags

19 Feb 2015, 13:28
Can you please explain me how i resolve this equation? I cant seem to do it right.
its from the first example:

"Solution : The volume of the liquid is constant.
Initial volume = \pi * 5^2 * 9
New volume = \pi * r^2 * 4
\pi * 5^2 * 9 = \pi * r^2 * 4
r = (5*3)/2 = 7.5"

Kudos [?]: [0], given: 1

EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 10169

Kudos [?]: 3536 [0], given: 173

Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: 340 Q170 V170
Re: Math : 3-D Geometries [#permalink]

### Show Tags

19 Feb 2015, 19:56
Hi bogdanbb,

Your approach and solution are correct (the radius is 7.5). What part about it do you not understand?

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save \$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Kudos [?]: 3536 [0], given: 173

Intern
Joined: 10 Aug 2015
Posts: 31

Kudos [?]: 7 [1], given: 231

Location: India
GMAT 1: 700 Q48 V38
GPA: 3.5
WE: Consulting (Computer Software)
Re: Math : 3-D Geometries [#permalink]

### Show Tags

01 May 2016, 07:31
1
KUDOS
Hi bb,

None of the images are visible. Kindly check the image links. Thanks

Kudos [?]: 7 [1], given: 231

Founder
Joined: 04 Dec 2002
Posts: 15883

Kudos [?]: 29115 [0], given: 5272

Location: United States (WA)
GMAT 1: 750 Q49 V42
Re: Math : 3-D Geometries [#permalink]

### Show Tags

29 Jan 2017, 00:10
wings.ap wrote:
Hi bb,

None of the images are visible. Kindly check the image links. Thanks

Thank you for reporting this. It seems the hosted images that Shrouded uploaded were deleted by the host he used.
_________________

Founder of GMAT Club

Just starting out with GMAT? Start here... or use our Daily Study Plan

Co-author of the GMAT Club tests

Kudos [?]: 29115 [0], given: 5272

Re: Math : 3-D Geometries   [#permalink] 29 Jan 2017, 00:10
Display posts from previous: Sort by