How "Phi" Is My Face?     
by Suzanne Alejandre

Teacher Lesson Plan

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NCTM Standards:
Number and Operations:
      Understand numbers and relationships among numbers
      Understand and use ratios and proportions to represent quantitative relationships

      Understand processes of measurement
      Specify locations and describe spatial relationships using coordinate geometry

      Students use pixel measurements to calculate the ratios of different face measurements.
      Students determine how close a ratio is to the golden ratio or Phi.

Part I:

In the art world, if a face has certain measurements that are close to the golden ratio they are pleasing. This idea can be introduced to students in a variety of ways. The Web site, Phi: The Golden Number, has a lot of information on this topic.

There are many rectangles that could be used to illustrate this idea but the two that will be measured in this activity are illustrated here:

Top of head to chin.
Side to side.

Side to side at eye level.
Eye level to chin.

Students start the activity by opening the Java Applet Index page.

Note: It will open in a separate window. If you are displaying the page for students, arrange your browser windows so that the applet and the directions can be easily viewed. If students are working individually they should be encouraged to do this.

Ask the question:

Can you determine the "pixel" distance between two points?

Prompt students to notice the:

  • display of x (in pixels): How can you calculate how long a line segment is in pixels if you know the x coordinate of both endpoints of that line segment?

  • display of y (in pixels): How can you calculate how long a line segment is in pixels if you know the y coordinate of both endpoints of that line segment?

Students should be aware of the following:

If the line segment is horizontal, pay attention to the x-coordinates. If the line segment is vertical, pay attention to the y-coordinates.

Have the students work through the first example as a class, in groups, pairs or individually. Sample data is given for Annie's Face on the student page.

Part II:

Once students have the technique established, they can continue the activity by finding the measurements for all the photos and deciding which is the most perfect using the given criteria.


Ask students to generalize this method and write the steps needed to calculate the two different ratios.


A fun project would be to have the students' photographs instead of strangers' photographs. If you or your students have access to a digital camera, take photographs of the students.

Jill Britton tried this activity and wrote some excellent directions to follow. Please view: How to Replace the Photo in This Applet with Your Own.


Math Forum Resources

[Problems of the Week]  [T2T]  [Dr. Math]

[Math Forum Resources]

Selections from Math Forum Problems of the Week:

AlgPoW: The Length of Larry's Rectangle - posted January 3, 2000
Larry wants to know the length of his rectangle. Can you help him?

ElemPoW: Puzzling Puzzle Pieces - posted January 15, 2001
Explore the ways 108 puzzle pieces can be arranged.

[Math Forum Resources]

Selections from Teacher2Teacher Discussions:

The importance of teaching the Fibonacci Sequence, and creative ways to do so
What is the importance of teaching Middle Grades the Fibonacci Sequence? Do you know any NCTM standards that the sequence can relate to? Where can I find lesson plans on the Fibonacci Sequence for Middle Grades?

[Math Forum Resources]

Selections from Ask Dr. Math Archives:

Appearances of the Golden Number
Why does the irrational number phi = (1 + sqrt(5))/2 appear in so many biological and non-biological applications?

FAQ: Golden Ratio, Fibonacci Sequence
Please tell me about the Golden Ratio (or Golden Mean), the Golden Rectangle, and the relation between the Fibonacci Sequence and the Golden Ratio.

What is phi?

phi vs. Phi - a Coincidence?
Ancient and modern architecture reflect the 'golden ratio' (1.618. length to width) and this number is remarkably close to phi (.618...) seen in nature for leaf dispersions, etc. Is this just a coincidence?

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