# Lesson 15

Area of a Circle

## 15.1: Irrigating a Field (5 minutes)

### Warm-up

The purpose of this activity is for students to estimate the area of a circle by comparing it to a surrounding square.

### Launch

Explain that some farms have circular fields because they use center-pivot irrigation. If desired, display these images to familiarize students with the context.

Provide quiet think time followed by whole-group discussion.

### Student Facing

A circular field is set into a square with an 800 m side length. Estimate the field's area.

### Anticipated Misconceptions

Students might think the answer should be 640,000 m2 because that is the area of the square, not realizing that they are being asked to find the area of a circle. Ask them what shape is the region where the plants are growing.

Some students might incorrectly calculate the area of the square to be 6,400 m2 and therefore estimate that the circle would be about 5,000 m2.

Some students might try to use what they learned in the previous lessons about the relationship between the area of a circle and the area of a square with side length equal to the circle's radius. Point out that the question is asking for an estimate and answer choices all differ by a factor of 10.

### Activity Synthesis

Discuss the estimation strategies students used to answer the question. Ask students what the area of the square is in square meters ($$800 \boldcdot 800$$, or 640,000). Ask them if the circle's area is greater than or less than the square's area (less). Then ask them to use the picture to determine the best estimate (500,000 since the circle is close in area to the square).

## 15.2: Estimating Areas of Circles (20 minutes)

### Activity

In a previous lesson, students measured various circular objects and graphed the measurements to see that there appears to be a proportional relationship between the diameter and circumference of a circle. In this activity, students use a similar process to see that the relationship between the diameter and area of a circle is not proportional. This echoes the earlier exploration comparing the length of a diagonal of a square to the area of the square, which was also not proportional.

Each group estimates the area of one smaller circle and one larger circle. After estimating the area of their circles, students graph the class’s data on a coordinate plane to notice that the data points curve upward instead of making a straight line through the origin. Watch for students who use different methods for estimating the area of the circles (counting full and partial grid squares inside the circle, surrounding the circle with a square and removing full and partial grid squares) and invite them to share in the discussion.

For classes using the digital version, students can record the class data in the spreadsheet and graph points directly on the grid using the Point tool. Note: you have to click on the graph side of the applet for the point tool to appear.

Teacher Notes for IM 6–8 Accelerated
During the synthesis for this activity, create the table of diameter, estimated area, and area divided by diameter as suggested. Display the table for all to see throughout the lesson.

### Launch

Arrange students in groups of 2.

For classes using the print version, distribute the grids with the circles already drawn—one set of circles to each group of students from the Estimating Areas of Circles blackline master.

For classes using the digital version of the activity, assign each group of students a pair of diameters from this set:

• 2 cm and 16 cm
• 5 cm and 10 cm
• 3 cm and 12 cm
• 6 cm and 18 cm
• 4 cm and 20 cm
• 7 cm and 14 cm

Encourage students to look for strategies that will help them efficiently count the area of their assigned circles. Give students 4–5 minutes of group work time.

After students have finished estimating the areas of their circles, display the blank coordinate grid from the activity statement and have students plots points for their measurements. Give students 3–4 minutes of quiet work time, followed by whole-class discussion.

### Student Facing

1. Set the diameter of your assigned circle and use the applet to help estimate the area of the circle.

Note: to create a polygon, select the Polygon tool, and click on each vertex. End by clicking the first vertex again. For example, to draw triangle $$ABC$$, click on $$A$$-$$B$$-$$C$$-$$A$$.
2. Record the diameter in column $$D$$ and the corresponding area in column $$A$$ for your circles and others from your classmates.
3. In a previous lesson, you graphed the relationship between the diameter and circumference of a circle. How is this graph the same? How is it different?

### Launch

Arrange students in groups of 2.

For classes using the print version, distribute the grids with the circles already drawn—one set of circles to each group of students from the Estimating Areas of Circles blackline master.

For classes using the digital version of the activity, assign each group of students a pair of diameters from this set:

• 2 cm and 16 cm
• 5 cm and 10 cm
• 3 cm and 12 cm
• 6 cm and 18 cm
• 4 cm and 20 cm
• 7 cm and 14 cm

Encourage students to look for strategies that will help them efficiently count the area of their assigned circles. Give students 4–5 minutes of group work time.

After students have finished estimating the areas of their circles, display the blank coordinate grid from the activity statement and have students plots points for their measurements. Give students 3–4 minutes of quiet work time, followed by whole-class discussion.

### Student Facing

1. For each circle, use the squares on the graph paper to measure the diameter and estimate the area of the circle. Record your measurements in the table.

diameter (cm) estimated area (cm2)
2. Plot the values from the table on the class coordinate plane. Then plot the class’s data points on your coordinate plane.

3. In a previous lesson, you graphed the relationship between the diameter and circumference of a circle. How is this graph the same? How is it different?

### Student Facing

#### Are you ready for more?

How many circles of radius 1 unit can you fit inside each of the following so that they do not overlap?

1. a circle of radius 2 units?
2. a circle of radius 3 units?
3. a circle of radius 4 units?

If you get stuck, consider using coins or other circular objects.

### Anticipated Misconceptions

Some students might be unsure on how to count the squares around the border of the circle that are only partially included. Let them come up with their own idea, but if they need additional support, suggest that they round up to a whole square when it looks like half or more of the square is within the circle and round down to no square when it looks like less than half the square is within the circle.

### Activity Synthesis

Invite selected students to share their strategies for estimating the area of their circle.

Next, ask "Is the relationship between the diameter and the area of a circle a proportional relationship?" (No.) Invite students to explain their reasoning. (The points do not lie on a straight line through $$(0, 0)$$.)

To help students see and express that the relationship is not proportional, consider adding a column to the table of measurements to record the quotient of the area divided by the diameter. Here is a table of sample values.

diameter ($$\text{cm}$$) estimated area ($$\text{cm}^2$$) $$\text{area} \div \text{diameter}$$
2 3 1.5
3 7 2.3
4 12 3.0
5 19 3.8
6 27 4.5
7 38 5.4
10 78 7.8
12 108 9.0
14 147 10.5
16 200 12.5
18 250 13.9
20 312 15.6

Remind students that there is a proportional relationship between diameter and circumference, even though there is not between diameter and area. Recall that students saw the same phenomenon when they examined the relationship between the diagonal of a square and its perimeter (proportional) and the diagonal of a square and its area (not proportional).

Speaking, Listening: MLR7 Compare and Connect. Ask students to prepare a visual display that shows how they estimated the area of their circle. As they work, look for students with different strategies that overestimate or underestimate the area. As students investigate each other’s work, ask them to share what worked well in a particular approach. Listen for and amplify any comments that make the estimation of the area more precise. Then encourage students to make connections between the expressions and diagrams used to estimate the area of a circle. Listen for and amplify language students use to make sense of the area of a circle as the number of unit squares enclosed by the circle. This will foster students’ meta-awareness and support constructive conversations as they compare strategies for estimating the area of a circle and make connections between expressions and diagrams used to estimate the area of a circle. Design Principles(s): Cultivate conversation; Maximize meta-awareness

## 15.3: Making a Polygon out of a Circle (20 minutes)

### Activity

The purpose of this activity is for students to use what they know about finding the area of a parallelogram to develop the formula for the area of a circle. This activity builds on the work students did in grade 6 when cutting and rearranging shapes in order to calculate their areas. In this activity, students cut and rearrange parts of a circle to approximate a parallelogram. They see that the area of the parallelogram would be calculated by multiplying half of the circle’s circumference times its radius. Since students are not familiar with the process of writing proofs, it is necessary to walk them through writing the justification that uses the formula for the area of the parallelogram to develop the formula for the area of the circle.

The construction in this activity shows that the constructed parallelogram has a height at most the radius of the circle and a base at most half the circumference of the circle. Establishing equality is beyond grade level and will be addressed again in high school.

The process used to decompose the circle and recompose it into a shape resembling a parallelogram is a good example of MP8. The pie shaped wedges are successively cut in half and rearranged. Each time, the sides of the shape look more like line segments.

Watch for students who identify that the rearranged circle pieces resemble a parallelogram as the pieces get smaller. Also watch for how they estimate the width and height of this parallelogram and invite them to share during the discussion.

Teacher Notes for IM 6–8 Accelerated
This activity and the next each present a different method for informally deriving the formula for the area of a circle. You may select either method for arriving at the formula, but this activity is marked optional for the sake of time.

### Launch

Arrange students in groups of 2. Each group needs a circular object, with a diameter between 3 and 5 inches, and a thick marker with which to trace it. Also provide each group a sheet of white paper, a sheet of colored paper, a pair of scissors, and glue or tape. Remind students that in the past they decomposed and rearranged a shape to figure out its area. Demonstrate how to do the first 4 steps of the activity, and invite students to follow along with your example. Ask how the area of the new shape differs from that of the circle. Solicit some ideas on what the new shape resembles and how the area of such a shape could be approximated. Without resolving this, ask students to continue the process.

Engagement: Develop Effort and Persistence. Provide prompts, reminders, guides, rubrics, or checklists that focus on increasing the length of on-task orientation in the face of distractions. For example, provide students with a task checklist which makes all the required components of the visual display explicit.
Supports accessibility for: Attention; Social-emotional skills
Conversing, Reading: MLR2 Collect and Display. As students work in groups to make sense of the shapes glued on the paper, circulate and listen to the language students use as they compare the shapes and discuss how to find the area of the shape that resembles a parallelogram. Write down the words and phrases students use to explain why the areas of both shapes are equal and why the area of the shape is half of the circumference multiplied by the radius. As students review the language collected in the visual display, encourage students to clarify the meaning of a word or phrase. For example, a phrase such as “the base is half of the circle” can be clarified with the phrase, “the base of the parallelogram is half of the circumference of the circle.” A phrase such as, “the height is the radius” can be clarified with the phrase, “the height of the parallelogram is equal to the radius of the circle.” This routine will provide feedback to students in a way that supports sense-making while simultaneously increasing meta-awareness of language.
Design Principle(s): Support sense-making; Maximize meta-awareness

### Student Facing

Your teacher will give you a circular object, a marker, and two pieces of paper of different colors.

Follow these instructions to create a visual display:

1. Using a thick marker, trace your circle in two separate places on the same piece of paper.

2. Cut out both circles, cutting around the marker line.

3. Fold and cut one of the circles into fourths.

4. Arrange the fourths so that straight sides are next to each other, but the curved edges are alternately on top and on bottom. Pause here so your teacher can review your work.

5. Fold and cut the fourths in half to make eighths. Arrange the eighths next to each other, like you did with the fourths.

6. If your pieces are still large enough, repeat the previous step to make sixteenths.

7. Glue the remaining circle and the new shape onto a piece of paper that is a different color.

1. How do the areas of the two shapes compare?

2. What polygon does the shape made of the circle pieces most resemble?

3. How could you find the area of this polygon?

### Anticipated Misconceptions

Students might not fold the wedges accurately or make a straight cut. Remind them that the halves must be equal.

### Activity Synthesis

Ask students which polygon resembles the shape they constructed of circle pieces and how they know. (A parallelogram or rectangle)

For a dynamic visualization, see http://ggbm.at/RUqSMrjn, created in GeoGebra by Malin Christersson, or display this image.

Ask, “If we could continue cutting the wedges in half, how would that affect the new shape?” (It would look even more like a parallelogram or rectangle. The bumpy top and bottom straighten out, and the slanted height becomes more vertical.)

Note that we are going to refer to the bumpy parallelogram-ish shape as the “parallelogram” (in quotes). Ask students to describe comparisons we can make between the measurements in the circle and the “parallelogram.” Record and display these ideas for all to see. It may be helpful to write over the actual images themselves. Students should notice the following measurements, but if they do not, prompt them to look for them:

• The base of the “parallelogram” is approximately equal to half of the circle’s circumference.
• The height of the “parallelogram” is approximately equal to the radius of the circle.
• The areas of the 2 shapes are equal.

Tell students to label these measurements on their visual display:

• $$\text{Circumference} = \pi d$$” around the circle
• $$\frac12 \text{Circumference} = \pi r$$” at the base of the “parallelogram”
• “Radius” on the radius of the circle (needs to be drawn in)
• “Radius” on the height of the “parallelogram” (needs to be drawn in)

If students struggle to understand these relationships through the abstract variables, consider measuring your example circle and using the numerical measurements to talk about these relationships.

Ask students to discuss the different ways we can calculate the area of the “parallelogram.” Students should share the following ways:

• $$\text{Area} = \text{Base} \boldcdot \text{Height}$$
• $$\text{Area} = \frac12 \text{Circumference} \boldcdot \text{Radius}$$
• $$\text{A} = \pi r \boldcdot r$$
• $$\text{A} = \pi r^2$$

Tell students to record these ideas on their visual display. Display student work for all to see. If students do not bring up one of these ideas, make it explicit in the discussion.

## 15.4: Making Another Polygon out of a Circle (10 minutes)

### Optional activity

The purpose of this activity is for students to consider a different way to cut and reassemble a circle into something resembling a polygon in order to calculate its area. This time the polygon is a triangle, but the area of the circle can still be found by multiplying $$\frac12$$ times the circumference times the radius.

In the previous activity, students had experience following along as the teacher developed the justification. This time give students the opportunity to write their own justification for the area of a circle. As students work, monitor and select students who have clear, but different, explanations to share during the whole-group discussion. In particular, select students who use the following steps:

• $$\text{Area} = \frac12 \boldcdot \text{base} \boldcdot \text{height}$$
• $$\text{Area} = \frac12 \boldcdot \text{circumference} \boldcdot \text{radius}$$
• $$\text{Area} = \frac12 \boldcdot (\pi d) \boldcdot r$$
• $$\text{Area} = \pi r \boldcdot r$$
• $$\text{Area} = \pi r^2$$

If the bands making up the circle really did not stretch, then they would not form rectangles when they are unwound because the circumference of the inner circle is not the same as the circumference of the outer circle in each band. A rectangle is an appropriate approximation for the shape in terms of calculating its area.

Teacher Notes for IM 6–8 Accelerated
If the previous activity informally proving the area of a circle is not used, use this activity to informally derive the formula for area of a circle.

### Launch

Give students quiet work time followed by partner and whole-class discussion. Applet created in GeoGebra by timteachesmath.

Representation: Internalize Comprehension. Activate or supply background knowledge about finding the area of triangles and circles. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

Imagine a circle made of rings that can bend but not stretch.

Watch the animation.

1. What polygon does the new shape resemble?

2. How does the area of the polygon compare to the area of the circle?

3. How can you find the area of the polygon?

4. Show, in detailed steps, how you could find the polygon’s area in terms of the circle’s measurements. Show your thinking. Organize it so it can be followed by others.

5. After you finish, trade papers with a partner and check each other’s work. If you disagree, work to reach an agreement. Discuss:

• Do you agree or disagree with each step?

• Is there a way they can make the explanation clearer?

### Launch

Give students quiet work time followed by partner and whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge about finding the area of triangles and circles. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing

### Student Facing

Imagine a circle made of rings that can bend, but not stretch.

1. What polygon does the new shape resemble?
2. How does the area of the polygon compare to the area of the circle?

3. How can you find the area of the polygon?

4. Show, in detailed steps, how you could find the polygon’s area in terms of the circle’s measurements. Show your thinking. Organize it so it can be followed by others.

5. After you finish, trade papers with a partner and check each other’s work. If you disagree, work to reach an agreement. Discuss:

• Do you agree or disagree with each step?

• Is there a way to make the explanation clearer?

### Anticipated Misconceptions

If students struggle to imagine the circle and how it is cut and rearranged, suggest a familiar material for the rings that bends but does not stretch (for example, a cord or chain).

### Activity Synthesis

Ask selected students to explain their steps for finding the area in terms of the circle’s measures. Ask the class whether they agree, disagree, or have questions after each student shares their reasoning.

Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time. After students have the opportunity to think about how to find the area of the triangle in terms of the circle’s measurements, ask students to write a brief explanation of their process. As students prepare their explanation, look for students who state that the area of the shape that resembles a triangle is half of the circumference multiplied by the radius of the circle. Ask each student to meet with 2–3 other partners for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “Can you explain how…,” “You should expand on...,” etc.). Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students revise and refine both their ideas and their verbal and written output.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

The purpose of the discussion is to highlight that area and diameter (or radius) are not in a proportional relationship as well as reinforcing the formula for the area of a circle.

Ask students, “Is the relationship between the diameter and area of a circle a proportional relationship? How do you know?” (No, when we used estimates of the areas for circles of different diameters, it was clear that the points did not lie along a line.)

As students explain their reasoning, refer to the table created during the synthesis for the activity Estimating Areas of Circles to demonstrate that the relationship is nonproportional.

Here are some questions to help reinforce the formula for area of a circle:

• “How would you find the area of a circle with a radius of 10?” (Multiply $$\pi$$ and 100, because $$10^2 = 100$$.)
• “How would you find the area of a circle with a diameter of 10?” (Multiply $$\pi$$ and 25, because $$10 \div 2 = 5$$ and $$5^2 = 25$$.)
• “How would you find the area of a circle with a circumference of $$6\pi$$?” (First, find the radius by dividing the circumference by $$2\pi$$. For this circle, the radius is 3, because $$\frac{6\pi}{2\pi} = 3$$. Then we can use the formula to find the area. Because $$3^2 = 9$$, the area of the circle is $$9\pi$$ or approximately 28.26.)

## Student Lesson Summary

### Student Facing

The circumference $$C$$ of a circle is proportional to the diameter $$d$$, and we can write this relationship as $$C = \pi d$$. The circumference is also proportional to the radius of the circle, and the constant of proportionality is $$2 \boldcdot \pi$$ because the diameter is twice as long as the radius, so $$C=2\pi r$$. However, the area of a circle is not proportional to the diameter (or the radius).

The area of a circle can be found by taking the product of half the circumference and the radius. If $$A$$ is the area of the circle, this gives the equation:

$$A = \frac12 (2\pi r) \boldcdot r$$

This equation can be rewritten as:

$$A=\pi r^2$$

(Remember that when we have $$r \boldcdot r$$ we can write $$r^2$$ and we can say “$$r$$ squared.”)

This means that if we know the radius, we can find the area. For example, if a circle has radius 10 cm, then the area is about $$(3.14) \boldcdot 100$$ which is 314 cm2.

If we know the diameter, we can figure out the radius, and then we can find the area. For example, if a circle has a diameter of 30 ft, then the radius is 15 ft, and the area is about $$(3.14) \boldcdot 225$$ which is approximately 707 ft2.