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Discovering the Area Formula for Circles

  • Lesson
6-8
2
Measurement
Christopher Johnston
location: unknown

Using a circle that has been divided into congruent sectors, students will discover the area formula by using their knowledge of parallelograms. Students will then calculate the area of various flat circular objects that they have brought to school. Finally, students will investigate various strategies for estimating the area of circles.

Prior to the lesson, ask students to bring in several flat, circular objects that they wish to measure with their classmates.

As a warm-up, give students an opportunity to estimate the area of the circular objects that they have brought to class. Working in groups and using the Area of Circles Activity Sheet, students should individually complete the first two columns:

  • Description of the object
  • Their estimate for the area of the object

(The other two columns will be completed later in the lesson.)

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Area of Circles Activity Sheet

Students may use any method they like to estimate the area of their objects. Some possible methods include:

  • Students can trace the shape of their object on a piece of centimeter grid paper and count how many square centimeters make up the total area of the circle.
  • Students can divide the circle into wedges by drawing various radii. They can approximate the area of each wedge using the triangle formula. (This method is similar to a method used by Archimedes, and it is the method that will be used later in this lesson. For a connection to mathematical history, you may want to include a brief overview of Archimedes and his method for calculating the area of a circle.)
  • Students can inscribe the circle in a square, hexagon, or some other polygon. Then, the same shape could be inscribed within the circle. Students could determine the area of the inscribed and circumscribed shapes to get lower and upper estimates, respectively. (You may need to provide a sample drawing of this method, like the one shown below.)

    1852 inscircsquare 

    After students have estimated the area of several objects, allow them to physically discover the area formula of a circle. Since this is a whole-class activity, you may wish to enlarge the manipulatives and display them on the chalkboard, or you can use them on the overhead projector.

Distribute the Fraction Circles Activity Sheet.

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Fraction Circles Activity Sheet

Have students cut the circle from the sheet and divide it into four wedges. (This can be done if students cut only along the solid black lines.) Then, have students arrange the shapes so that the points of the wedges alternately point up and down, as shown below:

1852 four wedges 

Ask, "When arranged in this way, do the pieces look like any shape you know?" Students will likely suggest that the shape is unfamiliar.

Then, have students divide each wedge into two thinner wedges so that there are eight wedges total. (This can be done if students cut only along the thicker dashed lines.) Again, have students arrange the shapes alternately up and down. Again ask if this arrangement looks like a shape they know. This time, students will be more likely to suggest that the arrangement looks a little like a parallelogram.

Finally, have students divide each wedge into two thinner wedges so that there are sixteen wedges total. (This can be done if students cut along all of the dashed lines.) Allow students to arrange the wedges so that they alternately point up and down, as shown below:

1852 sixteen wedges 

Ask, "When the circle is divided into wedges and arrange like this, does it look like another shape you know? What do you think would happen if we kept dividing the wedges and arranging them like this?" Lead the discussion so students realize the shape currently resembles a parallelogram, but as it is continually divided, it will more closely resemble a rectangle .

You may wish to continue this activity by having students divide the wedges even further.

Ask students, "What are the dimensions of the rectangle that is formed?" From the Circumference lesson, students should realize that the length of the rectangle is equal to half the circumference of the circle, or πr. Additionally, it should be obvious that the height of this rectangle is equal to the radius of the circle, r. Consequently, the area of this rectangle is πr × r = πr2. Because this rectangle is equal in area to the original circle, this activity gives the area formula for a circle:

A = πr2

The figure below shows how the dimensions lead to the area formula.

1852 wedges with measure 

Allow students to return to the objects for which they estimated the area at the beginning of class. They should measure the radius of each object and record it in the third column on the Area of Circles sheet. Then, students should use the formula just discovered, calculate the actual area of each object, and record the area in the fourth column.

Once all groups have completed the measurements and calculations, a whole-class discussion and presentation should follow. On the chalkboard, the presenter for each group should record the areas for the objects. The students should compare the results of each group and discuss the accuracy of the areas found.

The class should also compare their original estimates with the actual measurements. On their recording sheets, have them highlight the objects for which their estimates were very close to their actual. Using a few sentences, have the students explain (on the recording sheet) why some estimates were closer than others.

During the class discussion, the following are some key points to highlight:

  • Emphasize that 3.14 is only one approximation for π. Refer to the Circumference lesson, and discuss the various estimates that were found for π and what caused these variations. Also explain that there are other approximations, but typically 3.14 is used because it is accurate enough for most situations and it is easy to remember. If students are curious, other approximations for π are given on the Pi Approximation sheet.
  • The total area is almost always an approximation. Because the value of π can only be approximated, any time the area of a circle is stated without the π symbol, it must be an approximation. For instance, a circle with radius of 5 inches has an exact area of 25π in.2 and an approximate area of 78.54 in.2. You might wish to hold a "mock debate" with one student taking each position (yes, it’s always an exact value; no, it’s not an exact value) giving examples and reasons to justify their position.
  • Students should be able to calculate radius from diameter and diameter from radius. In particular, students should realize that d = 2r.
  • Students should understand the area formula as described in your curriculum. Slight variations are possible, so the version in your textbook, standards, or other materials may be different from the formula presented in this lesson.
  • Circular objects
  • Calculators
  • Scissors
  • Compasses
  • Rulers
  • Area of Circles activity sheet
  • Fraction Circles activity sheet
  • Centimeter grid paper on overhead transparencies
  • Blank copy paper

Assessments 

  1. Students can solve the following practice problem:
    • The radar screens used by air traffic controllers are circular. If the radius of the circle is 12 centimeters, what is the total area of the screen?
      [A =  pr2, so the area of the radar screen is approximately 3.14 × 122 ≈ 452.16 cm2.]
       
     
  2. Working in pairs or groups, have students locate manhole covers and other circles on or near the school grounds. Have students measure the diameter of these circles and then determine the area.

Extensions
1. Students can use the Internet to research various methods for approximating the area of circles throughout history. In pairs, students could try the various methods and determine the accuracy of their results as compared to the formula that they found. What cultures used good methods that produced accurate results? Did anything surprise you about these methods or the results? Each pair of students could report back to the class using a poster, overhead transparencies, or PowerPoint presentation.
2. Using the Internet, students should find out the dimensions of a typical dartboard and the sizes of each point value sector. Using their knowledge of the area of circles, they can calculate the probability of hitting a certain point value. (Depending on the information that they find, students may need to estimate the area of certain sectors to find an approximate probability.)

Questions for Students 

1. In your opinion, why did we use the properties of a parallelogram to discover the area formula for circles?
 

[Determining the area of a circle is difficult. By converting a circle to a parallelogram, we can use the formula for the area of a parallelogram to determine the area of the circle.]

2. When would it be necessary to know the exact area of a circle? When would an estimate be sufficient? Explain your thinking.

[Student responses may vary.]

3.Why did we approximate our answers for area? Can the area of a circle ever be exact?

[It is not possible to find an exact numeric value for π. Therefore, all calculations of area must be approximations (unless the answer is left in "exact form," which means using the symbol π to express the answer).]

Teacher Reflection 

  • When students were working in pairs to find the area of their assigned circular objects, how precise were the students’ measurements and area calculations? When the results were discussed as a class, did those students who were not as precise while measuring demonstrate an understanding of how to get more precise measurements? Or did all students get basically the same results?
  • Did students use both metric and customary units of measure? With which were they more comfortable, and would future measurement lessons make them comfortable with the other?
  • Were concepts presented too abstractly? Too concretely? How would you change the presentation if this lesson were taught again?
  • How do you know that students were actively engaged in the learning process?
  • What content areas did you integrate within the lesson? Was this integration appropriate and successful?
  • Did you find it necessary to make adjustments while teaching the lesson? If so, what adjustments, and were these adjustments effective?
 
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Measurement

The Ratio of Circumference to Diameter

6-8
Students measure the circumference and diameter of circular objects. They calculate the ratio of circumference to diameter for each object in an attempt to identify the value of pi and the circumference formula.

Learning Objectives

Students will:
  • Measure the radius and diameter of various circular objects using appropriate units of measurement
  • Discover the formula for the area of a circle
  • Estimate the area of circles using alternative methods
 

Common Core State Standards – Mathematics

Grade 6, Geometry

  • CCSS.Math.Content.6.G.A.1
    Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.