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A journey back to three dimensions this week.
You have a cube. The surface area of the cube is 150 cm^2. A "cross section" of the cube is a shape that you get when you cut the cube with a plane - sort of like slicing it. If you cut the cube with a plane that is parallel to one of its faces, you will get a square. Questions: What would be the perimeter of that square? What is the perimeter of the largest rectangle you can get as a cross section? Can you figure out how to get an equilateral triangle as a cross section? Extra: What's the area of the square? the rectangle? the biggest possible equilateral triangle? To make explaining this easier, I've provided a labelled cube below. ABCD is the front of the cube, while EFGH is the back. Be sure to explain your answers so that someone else might learn something from reading your solution.
 SolutionsThis was a fun foray into three dimensions! 105 of you must have thought so as well, as you got it right. 39 folks got it wrong, many of them stumbling on the equilateral triangle. 43 of the 96 folks who attempted the bonus got it right. The square isn't too tough, if you look at the surface area part closely. The rectangle gets a little trickier - noticing the angled cut is a big step. A number of people tackled it by deciding what the longest possible segment you could get was, and worked from there, which is a very good strategy. The triangle is even more difficult. You have to look at your cube in a way that you aren't really used to. Michael Roberts of Edison High School (NJ) drew a nice succession of pictures along with the text, "Rotate the cube 45 degrees forward and 45 degrees to the right of the left." What you end up with is a picture of the cube where you are looking straight at a vertex. This changes your perspective somewhat! Roger Dieterich III of Smoky Hill High School gave an interesting explanation of that part. He said "the way to get an equilateral triangle is to cut the cube parallel to one of the corners." Now, of course, you can't actually do this, but it does make some sense, doesn't it? The area of the equilateral triangle stumped a lot of people. I think this is partly because it is confusing to have this triangle stuck in the middle of everything. What I do in situations like this is to draw my triangle somewhere else, away from the messy stuff, and then figure out the area from there. That makes it a lot easier for me. Three folks who didn't have problem with the area of the triangle have their solutions included below. Amanda Woodruff and Beth McCabe of Wilburton High School give a very nice explanation of the problem, and used their knowledge of 30-60-90 triangles to figure out that area. One problem with a task like this one is that there is a lot of stuff included in the answer. When you're answering a long problem, try to make your solution easy to read. Alexandra Sowa of Archmere Academy does just this, presenting her answer for each part, immediately followed by an explanation. This makes it it a lot easier to figure out. And of course, what would a problem like this be without some good pictures? I've included the solution of Allen Hsu and Mike Sands of Nitschmann Middle School. They also provide a nice simple explanation for everything. A couple things of which to make a note. First, be careful when you are labeling your rectangle - EFDC won't work, since it crosses over itself. Order does matter. Also, when you get some numbers for answers, look at them and see if they make sense. If you get 20 for the square and 120 for the rectangle in this problem, that doesn't really seem reasonable, does it? Don't assume that all of the math you did is correct! Several people have equilateral triangles with larger areas than the rectangles. Look carefully. Always read your solution over before you send it to see if it makes sense. This is exactly the sort of thing you might catch. A list of all the people who got this problem right follows the highlighted solutions below, and most of the solutions are also available.
From: Alexandra Sowa
Grade: 9
School: Archmere Academy, Wilmington, Delaware
Cross Sections of a Cube
November 17-21, 1997
1. The perimeter of the square is 20 cm.
If the surface area of the cube is 150 sq.cm. then the surface area of one
face is 150/6=25 sq.cm. The square root of 25 is 5 so the length of each side
of the cube is 5cm. The perimeter of the "cross section" of the cube parallel
to a face is a square which is 5cm on each side. Perimeter= 4 times 5cm=20cm.
2. The perimeter of the largest rectangle is 10 plus 2 times the square
root of 50 or approximately 24cm.
The largest rectangle is made by a plane which "cuts" the cube at a 45 degree
angle (on a diagonal). One such rectangle is AECG. Using the Pythagorean
Theorem the length of the rectangle through the diagonal is the square root of
(5 squared plus 5 squared)=square root of 50. The length of each of the other
two sides of the rectangle is 5cm.
3. An equilateral triangle as cross section can be obtained by rotating by 45
degrees the plane that created the largest rectangle.
One such triangle would be Triangle EBG. Each side of this triangle
measures the square root of 50 sq.cm.
BONUS:
1. The area of the square is 25 sq.cm.(5cm x 5cm).
2. The area of the rectangle is approximately 35sq.cm. (5cm x
the square root of 50cm).
3. The area of the biggest equilateral triangle is approximately
21.6 sq.cm. The height of the equilateral triangle of side
"square root of 50 sq.cm." is calculated, using the Pythagorean Theorem,
to be the square root of "50 minus 1/2 the square root of 50 squared" = (50
- 50/4)^1/2 = 37.5^1/2 or approximately 6.12 cm. 1/2bh= 1/2 x 50^1/2 x 6.12
or approximately 21.6 sq.cm.
Alexandra Sowa
AleCat6213
From: Allen Hsu and Mike Sands
Grade:
School: Nitschmann Middle School, Bethlehem, Pennsylvania
Subject: Geometry problem of the week November 17 - 21
This is from Allen Hsu and Mike Sands at Nitschmann Middle School
in Bethlehem, Pennsylvania.
All sides are edges are equal, therefore all diagonals are equal, as also the
area of each face of the cube.
The following students submitted correct solutions this week:Artem Dmytrenko, Grade 10, Immaculate Conception High SchoolGregory Pack, Grade Teacher, Northview High School, Bratt, Florida Gordon Bockus Jr, Grade Freshman, Eastern Oklahoma State College, Wilburton, Oklahoma Sean Kelly, Grade 7, Odle Middle School, Bellevue, Washington Roger Dieterich III, Grade 10, Smoky Hill High School, Aurora, Colorado Lev Navarre, Grade 6, Odle Middle School, Bellevue, Washington Mary Hanna, Grade 11, Myers Park High, Charlotte, North Carolina John Kuendig, Grade 9, Forest High School, Ocala, Florida Tracy Steed, Grade 12, Wilburton High School, Wilburton, Oklahoma Julia Le, Grade 11, Minnechaug Regional High School, Wilbraham, Massachusetts Avrum Tilman, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Joel Walker, Grade 8, South Fremantle Senior High School, Fremantle, Australia Laurence Troy, Grade 8, South Fremantle Senior High School, Fremantle, Australia Jackie Evans, Grade 9, Smoky Hill High School, Aurora, Colorado Bryan Baker, Grade 9, Smoky Hill High School, Aurora, Colorado Catherine Mangasi, Grade 12, Wilburton High School, Wilburton, Oklahoma Shawnelle Kelley, Grade 10, Shelby County High School, Columbiana, Alabama Jim and John, Grade 9, Franklin County High School, Rocky Mount, Virginia Ryan McGauley, Grade 8, Tappen Middle School, Ann Arbor, Michigan Bassel Rifai, Grade 10, Green Valley High School, Henderson, Nevada Tiffanie Lam, Grade 8, Sequoia Middle School, Pleasant Hill, California Jeffrey Wong, Grade 8, St. Ann School, Wollaston, Massachusetts Jon Gantman, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Kaitlin Primavera , Grade 10, Mount Saint Joseph Academy, Flourtown, Pennsylvania Katie Wright and Jessica Sink, Grade 9, Franklin County High School, Rocky Mount, Virginia Brian Thompson, Grade 9, Franklin County High School, Rocky Mount, Virginia Tammy Davis, Grade 11, Shelby County High School, Columbiana, Alabama Robin Thornburg, Grade 11, Shelby County High School, Columbiana, Alabama Noam Abrams, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvani Kevin Yurkerwich, Grade , Okemo Mountain School, Stowe, Vermont Alex Chen, Grade 7, Odle Middle School, Bellevue, Washington Matt Simcox, Grade 9, East Mecklenburg High School, Charlotte, North Carolina Stephanie Lareau, Grade 9, East Mecklenburg High School, Charlotte, North Carolina Megan Ross, Grade 9, East Mecklenburg High School, Charlotte, Nort Carolina Amanda Woodruff and Beth McCabe, Grade 11, Wilburton High School, Wilburton, Oklahoma Alisha Becker, Grade 10, Smoky Hill High School, Aurora, Colorado Jennifer Liang, Grade 8, Odle Middle School, Bellevue, Washington Anna Wu, Grade 10, Monte Sant' Angelo Mercy College, Sydney, Australia William Chin, Grade 11, Randolph Junior/Senior High School, Randolph, Massachusetts Alex Morgovsky, Grade 11, Akiba Hebrew Academy, Merion, Pennsylvania Xiaochang Li, Grade 8, Odle Middle School, Bellevue, Washington Alexandra Sowa, Grade 9, Archmere Academy, Wilmington, Delaware Aaron Sawyer, Grade 9, Rockford High School, Rockford, Michigan Robert Witting, Grade 9, Granada High School, Livermore, California Jenny Lurie, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Jenny Kaplan, Grade 7, Castilleja Middle School, Palo Alto, California Patrick Tartar, Grade 8, Odle Middle School, Bellevue, Washington Julia Fischer, Grade 10, Granada High School, Livermore, California Ashley Monroe, Grade 9, Casady School, Oklahoma City, Oklahoma Frannie Laks, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Jane Milton and Sara FitzSimmons, Grade 10, Mount Saint Joesph Academy, Flourtown, Pennsylvania Andrew Davis, Grade 9, Skyview High School, Vancouver, Washington Sonia Teas, Grade 10, Oak Park and River Forest High School, Oak Park, Illinois Jason Chiu, Grade 9, Laramie Junior High School, Laramie, Wyoming Ita Kristiana, Grade 8, South Fremantle Senior High School, Fremantle, Australia Mark Beevers, Grade 8, South Fremantle Senior High School, Fremantle, Australia Denny Chao, Grade 10, Germantown Academy, Fort Washington, Pennsylvania Marc Horton, Grade 8, Odle Middle School, Bellevue, Washington Will Marrs, Grade 9, Granada High School, Livermore, California Alison Miller, Grade 6, homeschooled, Niskayuna, New York Christine Heyer, Grade 8, Odle Middle School, Bellevue, Washington Lauren Rossi and Anne Hines, Grade 10, Germantown Academy, Fort Washington, Pennsylvania Ryan Smith, Grade 9, Germantown Academy, Fort Washington, Pennsylvania Thomas Kuo, Grade 10, Burroughs High School, Ridgecrest, California Helen Wong, Grade , Odle Middle School, Bellevue, Washington Andrew Cooledge, Grade 7, Odle Middle School, Bellevue, Washington Clayton Dillaway, Grade 8, Odle Middle School, Bellevue, Washington Zach Dillon, Grade 7, Odle Middle School, Bellevue, Washington Brandon Gilchrist and Krystal Schumann, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Kelsey Long and Emily Buzicky and Alison Falkenhagen, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Thuy Nguyen, Grade 11, Highland Park Senior High School, St. Paul, Minnesota Libbie Gies, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Eric Collins, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Lisa Oakland, Grade , Redmond High School, Redmond, Oregon Nathan, Grade , Redmond High School, Redmond, Oregon Amit Bhatia, Grade , Shameica Edwards, Grade 12, Wingate High School, Brooklyn, New York David Donlan, Grade , Redmond High School, Redmond, Oregon Allen Hsu and Mike Sands, Grade , Nitschmann Middle School, Bethlehem, Pennsylvania Matt Niederst, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Lydia Wang, Grade 9, Smoky Hill High School, Aurora, Colorado Paul Tarasi, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania Laura Roos, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania Aleksandra Kukic, Grade 10, Grand Park High School, Winnipeg, Manitoba, Canada Neil Seifried, Grade 10, Pullman High School, Pullman, Washington Amy Tappe, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Jen Erhart, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Niki Weber, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Lauren Moser, Grade , Shaler Area High School, Pittsburgh, Pennsylvania JIll Bisceglia, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Brian Bailey, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Rick Gazica, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Justin Dembowski, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Katie Schill, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania Jill Filipovitz, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Katie Beranek, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania Erin Roll, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Megan Bray, Grade , Shaler Area High School, Pittsburgh, Pennsylvania Lindsay Bertram, Grade 11, Fairfield High School, Fairfield, Connecticut Billy Ng, Grade , Grant Park High School, Winnipeg, Manitoba, Canada Alex Chernyavsky, Grade , Akiba Hebrew Academy, Merion, Pennsylvania Kristen Allegue, Grade , Ethel Walker School, Simsbury, Connecticut Zimran Douglas, Grade 11, Wingate High School, Brooklyn, New York Michael Roberts, Grade 9, Edison High School, Edison, New Jersey Najeh Adib and Devon Bertram and Rose Blackburn and Liz Carter and Kristen Eaker and Samantha Gen and Katie Greer and Emily Griffin and Danielle Haskard and Michael Klaff and Muyshann Ky and Cole Lulka and Jen McKnight and Jason Pollock and Briana Porco and Meryl Rosten and Erin Ryan and Sarah Taylor and Mi, Grade 11, Fairfield High School, Fairfield, Connecticut |
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The perimeter of the square is 20 because each side length is 5 because
sqrt(150/6) is equal to five.
The largest rectangle can can only be constructed by cutting along the
diagonals. Since each face is a square and the length of the diagonal are
longer than the side. Therefor the length of the diagonal is 5sqrt2 therefore
longer making the perimeter 10 + 10sqrt2
The equilateral triangle is created by cutting on the diagonals of three sides.
The sides are equal due to the fact that this is a cube therefore all the
diagonals must be equal also this creates the largest equilateral triangle. The
perimeter of this triangle is 15sqrt2 .
Extra: Area of square = 25
Area of the Rectangle = 25sqrt2
Area of Largest possible Equilateral triangle is (25sqrt3)/2