# Area of Circles

This lesson is based from the textbook Holt Middle School Math:  Course 2.
Bennet, J.M., Chard, D.J., Jackson, A., Milgram, J., Scheer, J.K., Waits, B.K.  (2004).  Holt middle school math:  Course 2.  Orlando:  A Harcourt Education Company.

Lesson Title:  Finding Areas
Course Title:  Compacted Math
Time Allotted:  1 class period
Number of Students:
24-34 students
None
Day 5

Goals and Objectives:

According to the NCTM Principles and Standards of Mathematics, the following standards are met in this lesson:
1.   To understand measurable attributes of objects and the units, systems, and processes of measurement.

• Understand both metric and customary systems of measurement.
• Understand relationships among units and convert from one unit to another within the same system.
• Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.

2.   To apply appropriate techniques, tools, and formulas to determine measurements.

• 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.

According to the NCSCOS, the following standards are met in this lesson:

1.   Competency Goal 2:  The learner will demonstrate an understanding and use of the properties and relationships in geometry, and standard units of metric and customary measurement.

Goals:

• For the students to understand through construction how to find the area of a circle.
• For the students to see the relationships between the area and circumference of a circle.

Materials Needed and/or Use of Space:

• Calculators
• Rulers
• Cut out circles for each group with diameter of about 12 inches
• Markers/Crayons
Math Fact of the Day:

What do you call a crushed angle?
A rect-angle

• Share this with the students at the beginning of class.  It is similar to when an English teacher shares a quote at the beginning of language arts class!!

Problem of the Day:

A 16 in. pizza sells for \$11.99.  A 10 in. pizza sells for \$5.99.  Which size gives you more pizza per penny?  Explain.

• Give this to the students at the beginning of class and have them turn in their answers with their unit portfolio at the end of the unit.  The answer for this problem is:  The larger pizza because you get more than twice as much (64pi square inches instead of 25 pi square inches) for about twice the price.
• This problem is an introduction into the day's lesson.
• The students should be able to solve this problem after today's lesson.

*NOTE:  Throughout this lesson, I use pi to symbolize 3.14 and the ^ sign is used to show that something is raise to a power.  For example, r^2 means "r squared".

Motivational Activity:

The students will learn about how to find the area of circles today.
• To begin with, ask the students to describe what they would need to know to plan a pizza party for the class.
• Lead the students into a discussion about how the size of the pizzas ordered affects the number of pizzas needed to feed the class.  Discuss that you can find the best deal by comparing the unit costs per square inch of two differently sized pizzas if you can determine the area of each pizza, a circular figure.
• This is where our lesson for today comes in handy because we will be discussing how to find the area of a circle.

Lesson Procedure:

Yesterday we learned how to find the area of triangles and trapezoids.  Today we are going to discuss how to find the area of circles.  We are going to do this first by deriving the formula for the area of a circle ourselves.

This activity was found at http://www.education-world.com/a_tsl/archives/00-2/area_of_circle_doc.shtml.

The objective of this activity is for students to form a rectangle by partitioning a circle and relate A = bh to A = pi r^2.
• First, break students into groups of 3.  Do this by their last names today - all of the last names that begin with A are in a group, all of the last names that begin with B are in a group, and so on.  Make sure that there are no more than 3 students in each group.
• Discuss with students practical applications for finding the area of a circle. Explain the problems associated with partitioning a circle into unit squares to find its area. (It's not exact).  Give suggestions on how the area might be determined.
• Pass out the paper circles, scissors, rulers and markers/crayons.
• Have students draw a diameter (it does not need to be exact), and use two different colors to fill in the resulting semicircles.
• Instruct students to cut the circle in half along the diameter. Then have them cut each of the resulting semicircles into four equal sectors (They should know what a sector is, but remind them that a sector is a region enclosed by two radii and the arc joining their endpoints). There are now a total of eight pieces, four of each color.
• Ask students to assemble the eight pieces so that they form a shape which resembles a rectangle. Provide them with the hint that the same colors should not touch. (The resulting shape consists of sectors "pointing" in opposite directions, side by side).
• Ask for suggestions as to how to make the shape more like a rectangle. (This can be achieved by cutting each of the sectors in half, again).  Give them hints to try to arrive at this answer.
• Have students cut each of the sectors in half, once more, resulting in a total of 16 equal sectors, eight of each color. Solicit suggestions as to how to make the shape even more like a rectangle. (This can be achieved by cutting each of the sectors in half over and over again). Note: Do not allow students to create more than 16 sectors since they can become unmanageable.  This would get too out of control.
• Ask students to again assemble the sectors "pointing" in opposite directions, side by side. Make sure that none of the same colors are touching.
• Ask students to equate the parts of the approximated rectangle to the parts of the original circle. The remainder of the lesson involves the mathematical derivation of the formula for the area of a circle.
• The base, b, of the rectangle is equivalent to half of the circumference, C. The height, h, of the rectangle is equivalent to the radius, r, of the circle. Therefore, using the formula for the area of a rectangle, A = bh, we get b = C/2 and h = r.
• So the formula for the area of the circle is now A = C/2*r.
• However, we know that the circumference of a circle is equal to the diameter multiplied by pi (d*pi).  Thus, the formula can now be written as A = (d*pi)/2 *r.
• Since the diameter is the same thing as twice the radius (2r), the formula can now be written as A = 2r*pi /2*r.
• Simplifying this equation, we arrive at A = r*pi*r or A = pi*r^2 as the area of a circle.
• Great job!  We figured out the area of the circle ourselves.  Now let's look more into this formula and do a few examples together.

Here is a chart similar to those we looked at the past two days to help you understand how to find the area of a circle, along with a picture to help you learn.

AREA OF A CIRCLE
 The area A of a circle is the product of pi and the square of the circle's radius r. A=pi*r^2 *Make sure that the students are reminded that the variable r in this formula works as a placeholder for the actual value of the radius.  Since r is squared, you must also square the value.  We cannot forget to square our value r.

Example 1:  Find the area of the circle.

*NOTE:  Remind the students that pi is approximatley 3.14, thus, we will substitute 3.14 for pi when calculating the area for circles. We use the formula and substitute for r.
A=pi*r^2
A=(3.14)(3 m^2)
A=(3.14)(9 square meters)
A= 28.26 square meters.

Thus, the area of the circle is 28.26 square meters..
• Note:  Our units are in square units.  Make sure to point this out to students and to explain that because we do not have a specific unit of measurement to use, that we assume the measurement to be in units.

Example 2:  Find the area of the circle. *NOTE:  Be careful - we are given the diameter, not the radius.

Since the radius is 1/2 of the diameter, we can find the radius by r=d/2.
So, r=10/2=5 cm
Now let's find the area of the circle.

A=pi*r^2
A=(3.14)(5^2)
A=(3.14)(25)
A=78.5 square centimeters.

Thus, the area of our circle is 78.5 square centimeters.

Example 3:  There are circular fields located in the Wadi Rum Desert in Jordan.  On these fields, crops are grown on circular patches of irrigated land.  If the radius of one irrigated field is 60 feet, what is the area of the field?  Round your answer to the nearest whole number.

So, we know that r=60 feet.  Let's find the area!

A=pi*r^2
A=(3.14)(60^2)
A=11,304 square feet.

Thus, the area of the field is is about 11,304 square feet.

• Journal:  Students are asked to complete this journal entry for the day:  How do you think perimter and circumference are important in the real world?  How about finding the areas of parallelograms, areas of triangles and trapezoids, and areas or circles?  Can you think of any jobs where these mathematical concepts may be applied?  Please give at least 2 examples of where you could apply the new mathematical concepts that you have learned to the real world.  Be thorough in your examples and explain the applications of these concepts to the real world.

Extension:

If an extension activity is needed, the students can think and write responses to the following questions:

• Compare finding the area of a circle when given the radius with finding the area when given the diameter.  What is the difference?
• Explain how to find the area of a circle with a diameter of 3 feet.
• Give an example of a circular object in this classroom.  Tell how you could estimate the area of the object, and then estimate the are of the object.  You may use a ruler if needed.

Closure:

As you can see, we have learned how to find the area of circles.  We first derived this formula by starting with a circle and manipulating the circle to see how we could find the area of the circle.  We discussed at the beginning of class the example with the pizza and how we could find the better buy simply by finding the area of each pizza and comparing the prices.  This is something that you would use in your everyday lives.  Tomorrow we are going to discuss how we can apply the concepts that we have learned in the past week to the real world.

Assessment:

• Students will complete the problem of the day.  Their completion of this problem will show the teacher whether or not they understand what they have learned in class that day.  This activity also teaches the students more about mathematical communication.
• The students will also have a worksheet to complete.  If the worksheet is not completed in class, it is to be finished as homework.  This worksheet will count as a homework grade.
• The students are expected to participate in the activity, too.  This will be part of their classwork grade and their behavior/conduct grade.
• Students are also expected to complete the journal entry for today, too.  This is part of their journal/problem of the day grade.

Evaluation of Lesson Upon Completion:

• What would you do differently?
• What would you do the same?