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Modeling and Applications

In this category, each problem includes a reallife application of
one or more geometric concepts, or requires students to use a model
that they draw, build, or visualize. While many of the Geometry
Problems of the Week use the "real world" as a setting, these problems are
the most like "real" reallife scenarios.
Almost all of these problems are listed in other categories.
 Related Resources

Interactive resources from our Math Tools project:
Geometry: Modeling and Applications
The closest match in our Ask Dr. Math archives:
High School: Practical Geometry
High School: Polyhedra
NCTM Standards:
Geometry Standard for Grades 912
Access to these problems requires a Membership.

As Far as the Eye Can See
 Annie Fetter

Geometry, difficulty level 4. If I'm 5'10" tall, can I see the entire length of Lake Memphramagog, which is about 23 miles long?
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Assessing Seismic Risk
 Craig Foster and Annie Fetter

Geometry, difficulty level 3. Figure out how likely it would be for an earthquake to occur within
certain distances of a building.
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Boxing Up Harry's Broom
 Annie Fetter

Geometry, difficulty level 1. Help Harry find the right size box for his broom.
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Broken Pottery
 Annie Fetter

Geometry, difficulty level 2. Explain how to find the center of this sherd.
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Building a Rain Gauge
 Annie Fetter

Geometry, difficulty level 3. Given the depth of the water in a bottle, figure out how much rain really fell, considering that the opening of the bottle is much smaller than the main part.
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Building Bookshelves
 Steve Weimar

Geometry, difficulty level 2. Merriem and Emmanuel's new bookshelf hits the ceiling when they try to
put it in place. Explain why, and help them figure out how much to
cut off the bottom so that it will fit.
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Build Your Own Hexagon
 Annie Fetter

Geometry, difficulty level 2. Put a square on each edge of an equilateral triangle (edgelength one unit) and connect the outside vertices of adjacent squares to form a hexagon. Is this hexagon equilateral? equiangular? What is its area?
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Buying Paint and Shingles
 Annie Fetter

Geometry, difficulty level 2. Figure out how many gallons of paint and how many bundles of shingles I need to buy to finish my new shed.
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Calculating String Length
 Annie Fetter

Geometry, difficulty level 2. Given the archery targets of two competitors, figure out which one has the shortest 'string length'.
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Constructing a Window
 Annie Fetter

Geometry, difficulty level 4. Explain the shapes of the other four sides of a regular hexagonal window that's built between two wall studs 22.5" apart.
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Dendrochronology
 Annie Fetter

Geometry, difficulty level 2. Figure out about how old a large yellow birch is, given its diameter.
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Different Views
 Annie Fetter

Geometry, difficulty level 3. Given a front, top, and side view of an object, find the volume of that object.
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Field of View
 Annie Fetter

Geometry, difficulty level 2. Figure out the field of view for two different pairs of binoculars.
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Folding a Piece of Paper in Thirds
 Annie Fetter

Geometry, difficulty level 3. Show that the given folds mark a piece of paper exactly in thirds.
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Folding Postage Stamps
 Annie Fetter

Geometry, difficulty level 2. Fold a block of eight postage stamps, as indicated, along the perforations and without tearing, so as to form a single pile with the stamps numbered consecutively from top to bottom.
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Going Global
 Annie Fetter & NCTM's WLME

Geometry, difficulty level 2. Look at Shakespeare's Globe Theater from a mathematical standpoint  figure out the interior angle of an icosagon and calculate how many people could stand in the "field" in front of and around the stage.
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Is the Fair Fair?
 Annie Fetter

Geometry, difficulty level 3. Figure out the probability of winning at this popular midway game.
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Learning About GPS
 Annie Fetter

Geometry, difficulty level 4. What percentage of the surface of the earth can be seen by a satellite orbiting at an altitude of 20,200 km? Are the Global Positioning System satellites "geosynchronous"?
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The Length of a Belt
 Annie Fetter

Geometry, difficulty level 3. The diameters of two wheels are 18 cm and 12 cm and their centers are 30 cm apart. A belt goes around them and crosses 18 cm from the center of the big wheel. How long is the belt? How do you know you're right?
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Lost at Sea?
 Annie Fetter

Geometry, difficulty level 2. Help Darius and Skip figure out whether they're in the United States
or Canada using the bearings Skip takes.
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Lotsa Popcorn!
 Annie Fetter

Geometry, difficulty level 1. Can the World's Largest Popcorn Ball really be 50,000 times bigger than a
normal one? Wow!
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Magic Coins
 Annie Fetter

Geometry, difficulty level 3. How come you can fit a quarter through a hole the size of a dime? Find a pair of coins in another system of money that could be used as "magic coins"  the "biggest" and "smallest" coins for which this would work.
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Make a Square!
 Annie Fetter

Geometry, difficulty level 3. Make two cuts in the given figure and rearrange the pieces to make a square.
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Making a Map
 Annie Fetter

Geometry, difficulty level 3. Given a list of relationships between five different objects, explain which ones you could uniquely place on a map.
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Making Pie Crust
 Annie Fetter

Geometry, difficulty level 3. How would you change the recipe for a 9inch pie crust to make an 11inch pie crust?
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Measuring the Circumference of the Earth
 Annie Fetter

Geometry, difficulty level 3. At our office on the autumnal equinox, the angle formed by the sun's rays and a stick is 40 degrees. If our office is 2749 miles from the equator, what's the circumference of the Earth?
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The Mobius Band
 Annie Fetter

Geometry, difficulty level 2. Build a Mobius band and explain what happens when you cut it up.
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Mowing My Lawn
 Annie Fetter

Geometry, difficulty level 3. My lawn is 75 feet by 42 feet, my lawn mower cuts 18 inches at a time. I am mowing in a circular pattern, and have finished part of the lawn. That part took me 8 minutes. How much longer will I have to mow to finish the lawn?
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My Mom Rocks!
 Annie Fetter

Geometry, difficulty level 3. Help my mom decide if she's buried the stones in her new garden deep enough.
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Nautical Navigation
 Annie Fetter

Geometry, difficulty level 2. Help the sailors of the Kaleidoscope figure out what time
they'll pass the lighthouse and how far from the lighthouse they'll be
at that time.
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Nets of a Cube
 Annie Fetter

Geometry, difficulty level 2. When you fold up the given net of a cube, what shape is formed by the segments on the faces of the cube?
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Olympic Softball
 Annie Fetter

Geometry, difficulty level 2. Should the pitcher's rubber in Olympic softball be behind the line drawn from third base to first base, or in front of it?
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Painting a Cube
 Annie Fetter

Geometry, difficulty level 3. A large cube, made up of smaller cubes, has some of its faces painted. When it's taken apart, 24 of the small cubes have no paint on them. How big was the large cube, and how many of its faces were painted?
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Patterning a Shotgun
 Annie Fetter

Geometry, difficulty level 2. Construct a circular target with nine regions of equal area, with one a circle in the middle and spokes going out to the edge. If the whole circle is 6 feet in diameter, what will the diameter of the inner circle be?
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Pentomino Cubes
 Annie Fetter

Geometry, difficulty level 3. Given the 12 pentominoes, explain which ones can't be used as the basis for the net of a cube. Then take three of the ones that will work and explain where you can attach the sixth square to make the full net.
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Picking Painted Cubes
 Annie Fetter

Geometry, difficulty level 2. A large cube, composed of smaller cubes, is painted on all sides. The edge length is five small cubes. It's then broken apart and the smaller cubes are put in a bag. What are the odds that a chosen cube is painted on three sides? two? one? none?
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Playing with Blocks
 Annie Fetter

Geometry, difficulty level 4. A large cube is made up of 13 double cubes like the smaller one shown, plus a single cube. What color is the single cube, and where does it have to be?
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Playing with Matchsticks
 Annie Fetter

Geometry, difficulty level 2. Thirteen matchsticks are arranged to make six equal regions. Take away one of the matchsticks and arrange the remaining twelve so that they still make six equal regions.
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Playing with Paint
 Annie Fetter

Geometry, difficulty level 2. Explain what threedimensional solid would make a given "trail" if it were dipped in paint and rolled on the floor.
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Pondering Pantographs
 Annie Fetter and Lillian Ray

Geometry, difficulty level 3. Why does a pantograph work?
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Replacing the Football Field
 Annie Fetter

Geometry, difficulty level 2. Will a regulationsize NCAA soccer field fit inside an equalquadrant 400meter track?
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Running a Mile on a Metric Track
 Annie Fetter

Geometry, difficulty level 4. How far out from the rail on a 400 meter track would you have to run to cover 440 yards?
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Save the Lake!
 Annie & Tom Fetter

Geometry, difficulty level 2. How far does the water level drop if you take 900,000 gallons of water out of a lake with a surface area of 334 acres?
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Shotgun Targets
 Annie Fetter

Geometry, difficulty level 2. Figure out how large to make the inner circle of this shotgun target.
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Sighting in a Rifle
 Annie Fetter

Geometry, difficulty level 3. Analyze the given target of four shots, and tell me how far to move the scope on my rifle so that the group of shots will be centered around the bull'seye.
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The Speed Skating Oval
 Annie Fetter

Geometry, difficulty level 2. Find the radii of the turns of a 400 meter speed skating oval.
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Studying Stars
 Annie Fetter

Geometry, difficulty level 2. Given a number of points spaced around a circle, how many lines does it take to connect each point to all the other points except for the ones next to it?
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Taking Pictures of the Earth
 Annie Fetter

Geometry, difficulty level 4. If you wanted to go out into space and take a picture of the earth, how far from the earth would you have to go to make sure that you got 1/4 of the equator in your picture? How about 1/3?
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Voulez Vous des Voussoirs?
 Annie Fetter

Geometry, difficulty level 2. Voussoirs are trapezoidal chunks of stone used to build stone arches.
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Waiting for Hot Water
 Annie Fetter

Geometry, difficulty level 2. If a water pipe is 1/2" in diameter, the faucet is about 35 feet from the hot water tank, and the faucet runs at 3 gallons per minute, how long will it take the hot water in the tank to reach the faucet?
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Well, Well, Well
 Annie Fetter

Geometry, difficulty level 1. Figure out the diameter of my cousin Frank's new well.
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Where's Juanita Walking?
 Lillian Ray and Annie Fetter

Geometry, difficulty level 4. Help Juanita figure out how many possible ways she could walk to work.
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Who Wants to be an Archery Champion?
 Annie Fetter

Geometry, difficulty level 2. To determine the winner, calculate the "string length" for each of two archery rounds.
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Winning at Biathlon
 Annie Fetter

Geometry, difficulty level 3. Figure out which of two fictional biathletes will win a race.
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