HCPSS Worthwhile Math Task
Seven Wonders of the Geometric World Virtual Tour
Common Core Standard
G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a cylinder).
Common Core Traditional Pathway: Geometry, Unit 3
The Task
Welcome to the Seven Wonders of the Geometric World Virtual Tour. On this tour you will
experience some of the greatest man-made marvels from antiquity to the modern age. The tour
begins with the oldest structure on the tour and proceeds to the newest structure. At each stop on
the tour you will be asked to make observations and complete various calculations. You will use
your findings to compare and contrast the Seven Wonders of the Geometric World.
1. Great Pyramid of Giza (about 2,500 BC)
height = 449 ft
base side length = 745 ft
http://www.flickr.com/photos/yahl/4327051200/sizes/z/in/photostream/
2. Parthenon (432 BC)
height = 64 ft
length = 228 ft
width = 102 ft
http://www.flickr.com/photos/scatty/4414426311/sizes/m/in/photostream/
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HCPSS Worthwhile Math Task
3. Leaning Tower of Pisa (1372)
height = 187 ft
diameter = 51 ft
http://www.flickr.com/photos/irenetong/243426908/sizes/m/in/photostream/
4. Empire State Building (1931)
height = 1250 ft
length = 424 ft
width = 187 ft
http://www.flickr.com/photos/victoriapeckham/491273357/sizes/z/in/photostream/
5. US Department of Defense Headquarters “The Pentagon” (1942)
(assume the building is solid, this will account for below
ground levels)
height = 77 ft
side length = 921 ft
http://www.flickr.com/photos/gregwest98/3733687366/sizes/m/in/photostream/
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HCPSS Worthwhile Math Task
6. Air Force Academy Cadet Chapel
height = 150 ft
width = 84 ft
length = 280 ft
http://www.flickr.com/photos/paigehenderson/4214052860/sizes/z/in/photostream/
7. Spaceship Earth Epcot Center
circumference = 518ft
http://www.flickr.com/photos/aloha75/4806541657/sizes/z/in/photostream/
At each destination on the tour make the following observations and calculations:
A. Determine the 3-dimensional figure that the building most closely resembles.
B. Determine the 2-dimensional figure that is a horizontal cross section of the building
parallel to the base or through the center.
C. Calculate the volume of the building that is above ground.
D. Determine how many classrooms of the size of our classroom will fit inside of each
building on the tour.
E. Make any additional observations that about the building that are unique to the
building.
Once you have completed the tour prepare a presentation for the class detailing what you have
learned on the Seven Wonders of the Geometric World Virtual Tour. Be sure to make
connections or comparisons between the buildings and contrast differences between them.
Finally, determine the 3-dimensional figure that you believe is the best use of space when
constructing buildings. Be sure to explain your reasoning for choosing the 3-dimensional figure
using mathematics.
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HCPSS Worthwhile Math Task
Facilitator Notes
1. Divide the class into groups of 4’s. Working in their groups have the students complete
the Seven Wonders of the Geometric World Virtual Tour.
2. Once the groups complete the tour they should prepare a 5 minute presentation for the
class. If you have access to mobile labs the presentation could be done electronically and
projected using an LCD projector.
3. Following the presentations use the follow-up questions below to facilitate a class
discussion involving the volumes of three-dimensional.
4. The presentations should be evaluated based on correctly finding the volume, finding the
number of times the classroom would fit into a building, and the effort the group put into
making the presentation informative for the class. The comparisons, contrasts, and “best
use of space” questions are designed to be starting points for classroom discussions and
student discourse.
5. This activity could be done online with students searching for their own examples of
buildings that model 3-dimensional figures. Instead of giving the seven buildings ask the
groups to search the internet for examples of famous buildings that model a rectangular
prism, cylinder, sphere, square pyramid, etc.
Follow-Up Questions
1. What does the term “best use of space” mean to you?
2. How did your understanding of the meaning “best use of space” influence your choice of
the 3-dimensional figure?
3. What is the most commonly shaped building in cities? Why do you think that buildings
use this shape?
4. The Pentagon is the world’s largest office building in terms of floor area with 6,500,000
square feet. The Pentagon has 2 floors underground and 3 floors above ground. With
this in mind why do we not find more pentagonal shaped buildings in cities?
Solutions
Seven Wonders of the Geometric World Virtual Tour Key:
(Calculations are rounded to the nearest hundred)
1. Great Pyramid of Giza
 Right Square Pyramid
 Square
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HCPSS Worthwhile Math Task

Volume = 83,068,700 cu ft
2. Parthenon



Right Rectangular Prism
Rectangle
Volume = 1,488,400 cu ft
3. Leaning Tower of Pisa
 Right Square Pyramid
 Square
 Volume = 382,000 cu ft
4. Empire State Building
 Right Rectangular Prism
 Rectangle
 Volume = 99,110,000 cu ft
5. US Department of Defense Headquarters “The Pentagon”
 Right Pentagonal Prism
 Pentagon
 Volume = 112,403,400 cu ft
6. Air Force Academy Cadet Chapel
 Right Triangular Prism
 Isosceles Triangle
 Volume = 1,764,000 cu ft
7. Spaceship Earth Epcot Center
 Sphere
 Circle
 Volume = 2,352,000 cu ft
The comparisons and contrasts will vary depending on the student group. Likewise the selection
of the “best use of space” figure will vary depending on the student group. Some groups may
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HCPSS Worthwhile Math Task
choose Right Rectangular Prism because of its prevalence in major cities as skyscrapers. Other
groups may choose the Right Pentagonal Prism because of the large volume of the “Pentagon”.
It is possible that student groups may select any of the other 3-dimensional figures as long as
they can argue for its selection. Finally, the calculation of the number of classrooms that will fit
into each building will be dependent on the dimensions of your classroom.
Follow Up Questions:
1. What does the term “best use of space” mean to you?
Possible answers:
 The best use of space may indicate a design that is easily adaptable to
traditional office spaces with right angle corners. Buildings designed in this
manner would not have floor area that was “used” when including square or
rectangular shaped offices. Designs with bases that are circular or not
squares/rectangles may not be the “best use of space”.
 The best use of space might also indicate a design putting a premium on the
floor area of the base such as a pyramid.
2. How did your understanding of the meaning “best use of space” influence your choice of
the 3-dimensional figure?
Possible Answers:
 If we are given a scenario where a 100 ft tall building can be erected using a
base that cannot exceed a square with 100 ft sides then the following can be
deduced:
 A right square prism will have the greatest volume at 1,000,000
cubic feet
 Any other figure will have less volume as the base would need
to be inscribed into the square will 100 foot sides. For example
a right cylinder with these restrictions has a volume of 785,000
cubic feet and a right square pyramid has a volume of 333,333
cubic feet.
 Therefore if we want to maximize the space or volume that can be created in
an urban setting then a prism with a square or rectangular base becomes the
figure with the “best use of space”.
3. What is the most commonly shaped building in cities? Why do you think that buildings
use this shape?
Possible Answers:
 The most likely answer is right rectangular/square prisms.
 This figure has a base that that is easily adapted to the street design in major
cities.
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HCPSS Worthwhile Math Task

It has also been used frequently to design and erect some of the world’s tallest
buildings.
 Finally, the right rectangular/square prism is a figure that gives the office
building perhaps the most usable space.
4. The Pentagon is the world’s largest office building in terms of floor area with 6,500,000
square feet. The Pentagon has 2 floors underground and 3 floors above ground. With
this in mind why do we not find more pentagonal shaped buildings in cities?
Possible answers:
 A pentagonal building creates floor space that that may be non-usable since it
does not use right angles in the exterior walls.
 The Pentagon has a very large base area and is not very tall. This may be
impractical in a crowded city.
 Buildings in crowed cities need to use the space between streets. In cities
these regions between streets are often rectangles or squares.
 Another example of a pentagonal building is the World trade Center in
Baltimore. It has a much smaller base area, but has 30 floors.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License.