Thank you for your interest in NCTM’s Illuminations. Beginning in mid-April, all Illuminations content will be moving to Interactives will remain openly available and NCTM members will have access to all Illuminations lessons with new filtering and search options. We hope you will continue to utilize and enjoy these resources on

Pin it!
Google Plus

Looking for Patterns

Location: Unknown

Students discover Euler's Formula, a way of calculating the number of faces, edges, and vertices of geometric solids.

In this lesson, students refer to the table that they completed in the previous lesson, Study the Solids.

Return the Exploring Geometric Solids activity sheet to the students if you collected it in the previous lesson, as they will need it for today's lesson.

pdficon Exploring Geometric Solids Activity Sheet

The table introduces the students informally to Euler's Formula,

   faces + vertices = edges + 2,

which is often written as

   F + V - E = 2.

Students are asked to refer to the table they completed in the previous lesson to answer the following questions.

Look at the first shape in the table you made.

  • Find the sum of the number of faces and the number of vertices. How does this sum compare with the number of edges?
  • Do you think this may be a rule for the other shapes?
  • Add the number of faces and the number of corners for the other shapes in the table. Compare the sum of faces and corners to the number of edges. What did you find out? Is there a rule for all of the shapes?

Note that these questions appear on the activity sheet as questions 1, 2, and 3. 

You may wish to mention that a famous mathematician discovered this formula a long time ago. His name is Leohnard Euler (pronounced 'Oiler'). This rule is called Euler's Formula:

faces + vertices = edges + 2

You could mention to your students that someday some of them might discover an important mathematical pattern and have a formula named after them!

Assessment Option

Collect the students' activity sheets to assess student understanding. Answers to the activity sheet are available.


Move on to the next lesson, Construct a Solid.


Getting to Know the Solids

Students are introduced to some of the basic polyhedra. Students explore the shapes of the faces of these solids.

Study the Solids

In this interactive geometry investigation, students explore geometric solids and their properties. Specifically, students count the number of faces, edges, and corners (vertices) in various solids.

Construct a Solid

Students construct physical models of geometric solids.

Making A Shape Jacket

Students identify which geometric solids can be made from given nets. Students also create nets for common geometric solids.

Learning Objectives

Students will:

  • Analyze characteristics and properties of three dimensional geometric shapes and develop mathematical arguments about geometric relationships.
  • Use visualization, spatial reasoning, and geometric modeling to solve problems.
  • Discover Euler's Formula.

NCTM Standards and Expectations

  • Identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes.
  • Use geometric models to solve problems in other areas of mathematics, such as number and measurement.