Area of an Annulus: Discover and Apply

Most people know how to eat a doughnut.  But, do you know how to find the area of a doughnut?  Of course for the sake of argument, I'm talking about a flat dounghnut when we are working with area.  The amount of area between the outer circle and the inner circle provides us a unique look at the application of the area formula for a circle.

Engaging Questions

  • What do you call an annulus that is covered in sugar, chocolate, or maple and very tastey to eat?

  • What kind of annuls takes you to and from school every day?

Teacher Goals

  • The student will investigate the concept of area.

  • The student will derive the formula for calculating the area of an annulus.

  • The student will apply the area formula for an annulus.

Required Resources

  • Centimeter stick

    Some type of measuring device that has centimeters and/or millimeter.

  • Area Formula of a Circle: A = (pi)rr

    Use the Lesson "Area of a Circle: Discover and Apply" if you do not know the area formula for a circle.

  • No Description Available

Optional Resources

  • Calculator

Steps

  1. Question #1

    What is the area of the large circle with radius R?

  2. Question #2

    What is the area of the small circle with radius r?

  3. Describe It

    Write an algebraic equation to describe the process subtracting the small circle from the large circle.

  4. Practice Makes Perfect

    Practice working with the formula of an annulus using the attached worksheet.

  5. Application I: Tires

    I.  Go out to the parking lot and randomly identify 5 vehicles from which we will gather data.

    II.  Identify each vehicle by year, make, and model (1997, Mazda, Protege).
     
    III.  Identify brand of tire (Goodyear, Firestone, Douglas, etc.).
     
    IV.  Record the tire identification numbers found on the side of each tire (Ex: 205/75/R14)
  6. Application II: Calculate Annulus

    The sidewall portion of every tire is an annulus.  Calculate the annulus of one tire from each vehicle.  Leave your answers in centimeters squared.

Teacher Notes

  • Depending on the skill level of your students, this may take more than on day. Typically, this does not take much time to grasp the concept.

  • Cooperative learning is a good thing here. You might have the students work independently on deriving the formula. Then you might have them work in larger groups as they go calculate the area of the tires.

  • The Area of an Annulus is A = [(pi)R^2 - (pi)r^2], where A is the area, pi is from the greek alphabet for 3.14...., R is the radius of the large circle, and r is the radius of the small circle.

  • Duration: 1-2 class periods

Learner Notes

  • When calculating the area of a circle and rounding your result, do not round until the final answer. Also, if at all possible, use the (pi) button on your calculator instead of 3.14. This will help in a more accurate answer.

Mentor Notes

  • When calculating the area of a circle and rounding your result, do not round until the final answer. Also, if at all possible, use the (pi) button on your calculator instead of 3.14. This will help in a more accurate answer.

  • The Area of an Annulus is A = [(pi)R^2 - (pi)r^2], where A is the area, pi is from the greek alphabet for 3.14...., R is the radius of the large circle, and r is the radius of the small circle.

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