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Are They Possible?

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Students examine some isometric drawings that seem to be impossible and investigate one way Escher used to create these "impossible" figures.

In previous lessons, students saw that isometric drawings were not always what they appeared to be. A Dutch artist, M.C. Escher (1898-1972), is famous for his use of unusual perspectives to trick the viewer into seeing "Impossible Figures." In this lesson, students will examine some isometric drawings that seem to be impossible, and they will investigate one way Escher used to create these "impossible figures."

Project the following images for the students to see:

AreTheyPossible IMAGE EscherLikeConstructions 

As in previous lessons, you may print out the following PDF to create an overhead transparency:

overhead Are They Possible? 

Students should attempt to mentally construct each figure before using the isometric drawing tool.

appicon Isometric Drawing Tool 

Now, using the Isometric Drawing Tool, students should build each figure. Next, they should use the Inspect mode to look at the figure from different perspectives. Students can open several windows so they can have access to all three drawings.

Now students can try and create their own impossible figures. If students wish, they can print out their constructions using the Print function on the applet. As students work, circulate the room and use the Questions for Students section to engage students in math talk.
To wrap up the class, have students create one final construction. Have students print out their constructions and close their windows. This will prevent them from using the interactive to find the answers to the questions you will ask them. Ask students to:
  1. Find the surface area and volume.
  2. Sketch the front, right, and top (FRT) view.
  3. Sketch the mat plan.
  4. Discover if it is possible to have another figure that has the same isometric drawing.

You can either have students share their results or collect this as a form of assessment.


Assessment Options 

  1. Collect students' work as a form of assessment.
  2. Have students pair up and check each other's answers (for the last exercise of this unit). Circulate the room and take note of students who are struggling to remember previous lessons.


  1. Have students research M.C.Escher and his work. Have them write a paragraph on how isometric drawings are useful in art.
  2. A great lesson to follow up this unit is Hotel Snap, where students will be required to build a high-profit yielding hotel using snap cubes. This lesson will take approximately three days.

Questions for Students 

  1. Why do you think some people call these figures impossible? 
  2. What about isometric drawings creates these false impressions? 
  3. Do you think it is ever possible to have an isometric drawing that does not represent any 3-dimensional object? If so, can you draw one either on paper or using the applet? If not, can you explain why any isometric drawing created by the drawing tool is some 3D shape? 
Teacher Reflection 
  • Did this lesson wrap up the unit well? How else could you modify this lesson so that students see the relevance of isometric drawings?
  • Were students easily able to recall terms such as mat plan? If not, how could you review the material?
Unit Icon

Using Cubes and Isometric Drawings


Explore polyhedra using different representations and perspectives for three dimensional block figures. 

ExploringTheIsometricDrawingTool ICON

Exploring the Isometric Drawing Tool

Students explore using the isometric drawing tool and gain practice and experience in manipulating drawings.
FindingSurfaceAreaAndVolume ICON

Finding Surface Area and Volume

Using the isometric drawing tool, students build three-dimensional figures and find the surface area and volume of each figure.

Building Using the Front-Right-Top View

Students explore drawing the front-right-top view when given a three dimensional figure built from cubes. Students also explore building a three dimensional figure when given the front-right-top view.
MatPlans ICON

Mat Plans

Students explore drawing a mat plan when given a three dimensional figure built from cubes. Students also explore building a three dimensional figure when given the mat plan.
DoTheyMatch ICON

Do They Match?

Using three dimensional figures they have constructed, students determine when two isometric drawings can represent the same shape and explain their reasoning. Students will also determine what possible shapes might have the same isometric drawing and explain their reasoning.

Learning Objectives

Students will:
  • Examine isometric drawings that seem to be impossible.
  • Investigate one way Escher used to create these figures.

NCTM Standards and Expectations

  • Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Common Core State Standards – Mathematics

Grade 7, Geometry

  • CCSS.Math.Content.7.G.A.2
    Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.