Constructing in Three Dimensions: Math Task

  1. A pyramid is a polyhedron with a face called its base, which can be any polygon, and its other faces are all triangles meeting at a single point off of the base.   Here are two pyramids, one with a pentagon as base and the other with a rectangle as base:

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  1. Using your marshmallows and toothpicks, construct pyramids such that the triangular faces have sides which are one toothpick long.  Start with a pyramid having a three-sided (triangular) base, then construct one with a four-sided base, then five, etc., until you can no longer construct more pyramids.  For each pyramid you construct, record all information in the attached table.


  1. Eventually you were unable to make more pyramids.  Why was this case?


  1. Look at your table.  Do you notice any patterns related to the number of faces on a pyramid, the number of marshmallows needed to construct that pyramid, and the number of toothpicks needed?  Describe as many relationships as you can.


  1. There are many other polyhedrons besides pyramids.  One well-known polyhedron is the cube, which has 6 square faces:





a. Construct a cube with your marshmallows and toothpicks.  Record the number of faces, number of marshmallows used, and number of toothpicks used under the heading of “Cube and Other Polyhedron” in the table.


  1. How do the relationships between number of faces, marshmallows, and toothpicks for the cube compare to the relationships you listed in part 1. c.? 


Extension Question: 

Create another 6-sided polyhedron which is neither a cube nor a pyramid. Sketch the polyhedron below and/or describe it clearly with words. Did the number of marshmallows and toothpicks used fall in line with your earlier work? Can you describe a general rule which relates these attributes for all polyhedrons? You may wish to construct other polyhedrons and record their attributes in the table before you describe a pattern.


Pyramids (Problem 1)

Number of Sides of the Base

Number of Faces of the Solid Figure

Number of Marshmallows Used

Number of Toothpicks Used





























Cube and Other Polyhedron (Problem 2 and Extension)









© 2008 by Education Development Center, Inc. from The Fostering Geometric Thinking Toolkit. Portsmouth, NH: Heinemann. Reproduced with permission.