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Math : 3-D Geometries

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Math : 3-D Geometries [#permalink] New post 01 Oct 2010, 17:18
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3-D Geometries
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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

Image

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

Image

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

Image

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

Image

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

Image

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

Image

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
Answer is (d)

Example 2 : A spherical balloon has a volume of 972 \picubic 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
Answer (d)

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.

Answer is (d)

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

Answer is (c)

3-D Geometry Questions:

3-d-geometry-questions-171024.html

Some Other 3-D Problems

Sphere & Cube
Sphere & Cylinder
Cylinder & Cuboid
Cylinder & Cuboid II
Cylinder
Cube
Cube II
Cone
Cube III
Cylinder
Hemisphere
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Last edited by Bunuel on 28 May 2014, 03:17, edited 7 times in total.
added some examples
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Re: Math : 3-D Geometries [#permalink] New post 01 Oct 2010, 18:51
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Re: Math : 3-D Geometries [#permalink] New post 02 Oct 2010, 08:16
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Added some solved examples to the post
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Re: Math : 3-D Geometries [#permalink] New post 02 Oct 2010, 09:32
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Re: Math : 3-D Geometries [#permalink] New post 02 Oct 2010, 22:39
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WOOOOOOOOOOOOOOOOOOOOOOOOOOOOOWWWWWWWWWWWWWWWWWWWW

I sense Bunuel has competition!
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Re: Math : 3-D Geometries [#permalink] New post 02 Oct 2010, 23:09
Mathematica be the shizzles.
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Re: Math : 3-D Geometries [#permalink] New post 08 Oct 2010, 04:38
Thank you very much!
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Re: Math : 3-D Geometries [#permalink] New post 09 Oct 2010, 06:01
Kudos! for writing that up. 3-D is what was missing in GMAT Math Book, gotta update it in compiled GMAT Math Book pdf.
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Re: Math : 3-D Geometries [#permalink] New post 03 Jun 2011, 09:17
Thnks for sharing....
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Re: Math : 3-D Geometries [#permalink] New post 25 Aug 2012, 00:34
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
sphere.png [ 19.43 KiB | Viewed 8149 times ]


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Re: Math : 3-D Geometries [#permalink] New post 10 Jul 2013, 23:07
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Re: Math : 3-D Geometries [#permalink] New post 01 Jul 2014, 08:13
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
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Re: Math : 3-D Geometries [#permalink] New post 01 Jul 2014, 08:15
Expert's post
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


Writing Mathematical Formulas on the Forum: rules-for-posting-please-read-this-before-posting-133935.html#p1096628
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Re: Math : 3-D Geometries [#permalink] New post 01 Jul 2014, 08:27
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.

Answer is (d)



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 :)
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Re: Math : 3-D Geometries [#permalink] New post 01 Jul 2014, 09:24
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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.

Answer is (d)



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
faces-edges-vertices.png [ 13.61 KiB | Viewed 520 times ]


As for the question. Look at the image below:
Attachment:
MagicCube5x5.jpg
MagicCube5x5.jpg [ 71.65 KiB | Viewed 522 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
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Math : 3-D Geometries   [#permalink] 01 Jul 2014, 09:24
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