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Writing the Equation of a Circle

Michell Eike, Shayna Pond | Published: April 5th, 2022 by K20 Center

  • Grade Level Grade Level 9th, 10th
  • Subject Subject Mathematics
  • Course Course Geometry
  • Time Frame Time Frame 65 minutes
  • Duration More 1-2 class period(s)


In this geometry lesson, students will recall vocabulary about a circle, use their knowledge of midpoint and distance formulas to write the equation of a circle, and explore the connection between the equation of a circle and the Pythagorean Theorem. This is a multimodality lesson, which means it includes face-to-face, online, and hybrid versions of the lesson. The attachments also include a downloadable Common Cartridge file, which can be imported into a Learning Management System (LMS) such as Canvas or eKadence. The cartridge includes interactive student activities and teacher's notes.

Essential Question(s)

How are triangles and circles related?



Students recall what they know about circles.


Given points on a circle, students use their knowledge of the distance formula to find the radius and/or diameter.


Students complete guided notes with the class and/or watch a video to learn about the properties of a circle. These properties relate to the equation of a circle, how to write the equation of a circle given the endpoints of a diameter, and how to identify the center and radius from a given equation.


Students write the equation of a circle from a graph and investigate the relationship between the Pythagorean Theorem and the equation of a circle.


Students match equations with circles that have different characteristics.

Instructional Formats

The term "Multimodality" refers to the ability of a lesson to be offered in more than one modality (i.e. face-to-face, online, blended). This lesson has been designed to be offered in multiple formats, while still meeting the same standards and learning objectives. Though fundamentally the same lesson, you will notice that the different modalities may require the lesson to be approached differently. Select the modality that you are interested in to be taken to the section of the course designed for that form of instruction.



  • Lesson Slides (attached)

  • Circle Characteristics handout (attached; one per student; printed front only)

  • Guided Notes handout (attached; one per student; printed front/back)

  • Guided Notes (Teacher Guide and Model Notes) (attached; for teacher use)

  • Writing Equations handout (attached; one per student; printed front only)

  • Exit Ticket handout (attached; one half-sheet per student; printed front only)

  • Pencils

  • Coloring utensils (three per student)

  • Student devices with internet access (for Extend portion)



10 Minute(s)

Introduce the lesson using the attached Lesson Slides. Display slide 3 to show students the essential question. Go to slide 4 to share the lesson’s learning objectives. Review each of these slides with students to the extent you feel necessary.

Go to slide 5 and pass out the attached Circle Characteristics handout to each student. Ask students to use the Collective Brain Dump strategy to write what they know about circles on the top part of the handout. Have students use numbers and academic vocabulary to label the picture of the circle. Beside the picture, students should write anything else they know about circles that they did not label.

Then, ask students to pair up and compare their lists. Have students volunteer to share what they labeled, numerically and verbally. As students share out, write the circle characteristics on the board for the whole class to see.



10 Minute(s)

Display slide 6. Ask students to look at the Explore portion of the Circle Characteristics handout and work with their partners to complete it. Have students identify the center, radius, and diameter of a circle that has a center at (3, 5) and passes through (6, 9).



25 Minute(s)

Display slide 7 to provide students with the answers to the Explore portion of the Circle Characteristics handout.

Discuss students’ strategies for finding the dimensions of a circle without its graph. Come to an agreement about using the distance formula to find the radius.

Go to slide 8 and pass out the attached Guided Notes handout to each student. Complete the handout as a class. Once finished, have students add it to their math notebooks if that is a classroom norm.



15 Minute(s)

Display slide 9 and inform students it is time for them to apply what they have learned so far.

Pass out the attached Writing Equations handout to each student. Ask students to work in pairs to find the equation of each provided circle.

Transition through slides 10–11 so students can check their work. Be sure to allow students time to find their mistakes, ask questions, and refine their understanding.

Go to slide 12 and provide students with the link to the GeoGebra activity: Have students read about and interactively explore the relationship between the Pythagorean Theorem and the equation of a circle.



5 Minute(s)

Use the Exit Ticket strategy to assess students’ learning individually.

Go to slide 13 and give each student a half-sheet from the attached Exit Ticket handout. Then, ask students to match each of the following circles with one of the equations from the right-hand column of the Exit Ticket:

  1. A circle with a radius of 2 and a center at (–2, 3).

  2. A circle with a center at (2, –3) and passes though (2, –1).

  3. A circle with a given graph.

  4. A circle with endpoints of a diameter at (2, 7) and (2, –1).

Use student responses to see which misconceptions persist.