Illuminations: Geometry of Circles

Geometry of Circles
Using a MIRATM geometry tool, students determine the relationships between radius, diameter, circumference and area of a circle.

Learning Objectives
 
Students will:
  • Construct circles, and identify the diameters and centers of those circles
  • Understand the relationship between diameter and circumference
  • Understand the relationship between radius and the area

Materials
 
MIRATM Geometry Tool
Compass
Geometer's Sketchpad software program on the computer (optional)

Instructional Plan
Finding the circumference or area of a circle depends on the diameter of the circle. To help students develop an understanding of the characteristics of the diameter, have them construct a circle with a compass, and examine this circle with a a MIRA. A MIRA is a transparent geometry tool that reflects like a mirror. It can be used to bisect angles and segments or to explore geometric transformations. MIRAs are available from ETA Cuisenaire, Nasco, and other educational retailers.

 

 

As an alternative, hinged mirrors can be used for this lesson. Using simple paper folding can also work — after cutting out a circle, fold it in half, and the crease that forms is a diameter of the circle. However, MIRATM tools are definitely better.

Have students construct a circle using a compass. (Alternatively, you may wish to distribute a handout with circles already drawn.) Then, have them place the MIRA on the circle and explore; when one image maps onto the other, have them draw the MIRA line. Explain that the MIRA line is a diameter of the circle.

Then, allow students to construct several other MIRA lines for the same circle using the same process. Explain that each of these lines is a line of symmetry because each divides the circle exactly in half. Ask students, "How many lines of symmetry does a circle have?" [Infinite.] Explain that any of these lines of symmetry may be called a diameter, because each of them passes through the center of the circle.

Ask students, "How can you use the MIRA tool to find the center of the circle?" Give students a minute to do so. [The intersection of two diameters defines the center of the circle.] The segment from the center of the circle to the circumference is called the radius. What is the relationship between the radius and the diameter? [The radius is half the diameter.]

 

 

Have students construct another circle using a compass. Have students mark the spot where the point of the compass was placed. This is the center of the circle. Now, have students draw any chord of the circle. Have students use the MIRA to determine the perpendicular bisector of the chord. Through what special point does the perpendicular bisector pass? [The center.]

Allow students to work in pairs to complete either of the following constructions:

  • Draw a circle, and then construct an inscribed square so that the vertices of the square lie on the circumference of the circle. [One possibility is to use the MIRA to identify a diameter. Then use the MIRA to draw the perpendicular bisector of that diameter. The four points where the diameter and perpendicular bisector meet the circle are the vertices of an inscribed square.]
  • Inscribe a regular hexagon in a circle. [One solution is to draw any diameter, and divide it into two radii. Use the MIRA tool to draw the perpendicular bisectors of the two radii. Connect the four points where the bisectors intersect the circle with the two endpoints of the diameter, and a regular hexagon will be formed:

     

     

    An alternative solution is to use the length of a radius to mark off segments along the circle. Six congruent segments dictate a regular inscribed hexagon.]

Students may identify other solutions for these constructions. In addition, some students may have difficulty finding a solution for either construction. If that happens, allow students to struggle for a while. Eventually, however, you can have one group of students present their solution to the class; those students who had difficulty should then be asked to explain why the presented solution works.

Beyond these constructions, students need an understanding of diameter to examine the circumference of a circle. Allow students to investigate the ratio of circumference to diameter with the Circle Tool. Under the Intro tab of this tool, students are able to adjust the diameter of a circle, and they see that a little more than three copies of the diameter are needed to wrap entirely around the circle.

Circle Tool Circle Tool

Students can explore the relationship of circumference to diameter more explicitly. Under the Investigation tab, students can view various ratios in the table. By clicking the x/y button and selecting C as the numerator and d as the denominator, students will see circumference (C) in the first column of the table, diameter (d) in the second column, and the ratio of circumference to diameter (C/d) in the third column. By investigating circles of various size, students should see that the ratio of circumference to diameter is constant and has a value of approximately 3.14, or π.

Using this applet, lead students to see that C ÷ d = π, or C = π × d. Other ratios can be explored in a similar manner. For instance, the ratio of d/r can be explored in the table, and students should discern that the diameter is equal to twice the radius. This result then leads to another formula, C = 2πr. In the applet, if the ratio C/r were investigated, the result would be 6.28, or approximately 2π.

Finally, students can investigate the area of a circle by comparing it to the area of a square. If a circle is inscribed in a square, as shown above, the area of the square is 4r2, where r is the radius of the circle. Further, if a smaller square is then inscribed in the circle, its area is half the area of the larger square, or 2r2. This will lead students to guess that the area of the circle is approximately 3r2, which may cause some students to suspect that π may be involved. Such a conjecture can lead to a nice discussion and demonstration of the area formula, which can be conducted as described in the lesson Discovering the Area Formula for Circles.

Circle Tool Circle Tool

Upon completion of this lesson, students should understand the relationships between radius, diameter, circumference, and area.

Questions for Students
 
How do you know if a chord of a circle is also a diameter?
[If a chord is also a diameter, it will pass through the center of the circle.]
How is the diameter of a circle used to find its circumference?
[The value π represents the ratio of circumference to diameter of a circle. Consequently, C = πd, so the circumference can be found by multiplying the diameter by π.]
How is the radius of a circle used to find its area?
[The area of a circle with radius r is given by the formula A = πr2, so the area can be found by multiplying the radius squared by π.]

Assessment Options
 
  1. Provide each student with a circle of sufficient size, a ruler, and a calculator. The radius of each circle should be 3–10", and the circle should be constructed on heavy paper or cardboard. Do not give circles of the same size to each student. Students should use a ruler to perform any measurements and then determine the circumference. Once they complete their calculations, they should tell you their result. You should measure out a piece of string in the length that they request. Students should then glue the string to the circumference. (It is important that the teacher cut the string for this activity. If students are allowed to cut, they tend to continually cut the string until it fits the circle, rather than learning by doing.)

    Based on the results, students should explain what they discovered. Specifically, they should explain how well their calculations approximated the circumference and how close their string came to making exactly one revolution. Allow students to score their own work using the following rubric:

    • Advanced. My explanation went beyond the requirements of the task, with my reasoning communicated effectively. I was able to use logical reasoning to deduce relationships and test conjectures. Not only did my string fit exactly around my circle, but I used a systematic and logical process. I accurately measured the string to the proper degree of accuracy, using the correct formula and units.
    • Proficient. My string fit around the circle with no string left over and no gaps. I had a complete and correct solution process and explanation, using correct formula and units.
    • Basic. My string was either too long or too short. I did not use the correct formula or made computational errors, or my explanation was incomplete, unclear or unsystematic.
    • Unsatisfactory. I was unable to attempt this problem, or made an incomplete or incorrect attempt.

  2. Rather than revealing the area formula in class, allow students to write a conjecture about the formula. Students should explain their reasoning. Afterwards, a class discussion can be held; at the end of this discussion, the true formula can be revealed.

Extensions
 
  1. To develop understanding of the area of a circle, have pairs of students cut up a paper plate using lines of symmetry through the center, just as one slices a pizza. Rearrange the slices as shown below. Students will realize that this configuration almost looks like a rectangle! How would this "rectangle" help in finding the area of a circle? [The width of the rectangle is equal to the radius of the original circle. The length of the rectangle is half of the circumference, since the entire circumference is both on the top and bottom. Therefore, the area is equal to the radius times half the circumference, or A = ½Cr. Because C = =πd and d = 2r, this formula becomes the more familiar A = πr2.]

     

     

  2. Allow students to use Geometer’s Sketchpad or other geometry software to create the constructions described in this lesson.

  3. Research how hat sizes were determined! Or, check out the web site of a company that makes and sells hats, and you might find a table like the one below. What is the relationship between men’s head measurement (in inches) and American hat sizes? Have students measure the circumference of their head, and divide it by π — the result is their hat size.

     Hat Size 

     Head Circumference 
    (inches)

    6 3/8

    20 1/2

    6 1/2

    20 5/8

    6 5/8

    21

    6 3/4

    21 1/2

    6 7/8

    21 7/8

    7

    22 1/4

    7 1/8

    22 5/8

    7 1/4

    23

    7 3/8

    23 3/8

    7 1/2

    23 3/4

    7 5/8

    24

    7 3/4

    24 1/2

    If students were to plot these points in a scatterplot, one reasonable line of best fit is y = 3.14x, indicating that the y‑value (head circumference) is approximately π times the x‑value (hat size).

Teacher Reflection
 
  • Which parts of the lesson had high student enthusiasm? Low? Explain why this happened. How you could improve student enthusiasm if you teach this lesson again?
  • How do you know that students understood the material about circles? What did students do to demonstrate understanding?
  • What adjustments were needed while teaching this lesson? Were the adjustments successful?

NCTM Standards and Expectations
 
Geometry 6-8
  1. Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.
Measurement 6-8
  1. Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.
This lesson prepared by Rhonda Naylor.
  
1 period   

NCTM Resources

Navigating through Geometry in 6‑8

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