Lesson Plan

Measures of Arcs and Sectors

This lesson helps students to understand the connection between the arc as a fractional part of the circumference and the sector as a fractional part of the area of a circle.
Rafranz D.
School district administrator
Lufkin ISD
Lufkin, United States
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My Grades 5, 6, 7, 8, 9, 10, 11, 12
My Subjects Math

Students will be able to...

  • make conjectures about angles and arcs of circles and determine the validity of those conjectures.
  • Create and solve problems involving areas of sectors and arc lengths.
Grades 8 – 10
All Notes
Teacher Notes
Student Notes

1 Engage

Free to try, Paid


Prior Knowledge: Students should know that a circle has a degree measure of 360 degrees. They should also know how to determine the circumference of a circle as well as the area of a circle. In addition, algebraic determinations of the radius or diameter given the area or circumference would be helpful.

Hook: Have each student grab a circular coffee filter as they enter the roon and ask them to get it as flat as possible. 

Play this flocabulary video while kids are entering and flattening https://www.flocabulary.com/circles/


2 Explore

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Create a collaborative Padlet board for students to add "aha" moments and any connections related to the learning.

Start with developing vocabulary.Fold a coffee filter in half. The circle, and more specifically its arc, are divided into two equal parts. Ask kids about the degree of each half (semi circle) as well as each arc. How do you know?

In pairs, have kids fold the circle again, this time in fourths compare degrees of the central angle, arcs and areas. Make a conjecture about their relationship and justify.

Repeat the by folding once more, this time in eigths.

Record learning in padlet.

3 Explore/Explain


Have kids create a series of circles cut into various equal pieces (3,4, 5, 6, 8). Determine the degree measure of each arc. Students are to compare lengths of arcs to circumference and areas of sectors to areas of circles. Revisit conjectures from the coffee filter activity. Include a statement regarding conjectures and actual results as investigated in geogebra. Write a formula that can be used to determine the length of an arc or area of a circle given the degree of the arc.

Students can choose to create from the following options

  • One page showing all 5 circles with comparisons
  • One page showing all 5 circles as shown using hide/show or sliders

Students can choose to upload to geogebratube and share the link or save to google drive to turn in.

(Students may have questions about how to create circles divided into equal parts. This can be accomplished in multiple ways depending on the version of geogebra that students are using. The simplest way is to create a regular polygon and then a circle through three points on the polygon. Students would then hide the polygon and connect segments in the circle to form diameters. This method apprears via the chrome app but not the stand alone mac app)

4 Extend

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As the owner of a new pizza place, your job is to create a guide that customers can use to determine cost of various sizes of pizza given a certain radius and number of degrees. Using measures other than those demonstrated in the geogebra activity, make a smore flyer (or Tackk) that includes... 

  • At least 4 different pizza sizes by diameter including their areas and circumference
  • 2-3 examples of pizza slices accurately drawn by degree. (Geogebra is a great tool for doing this!)
    • You cannot use single slice degrees as demonstrated in the Guided practice.
    • Include areas of sectors and arc lengths for each.
  • Include the process for determining sector lengths and areas so that any person, regardless of math skill can follow your work.
  • Post the link to your kid blog as well as a reflective statement about what you learned.

When considering pricing, think about the degree or area and how those values might be used. 

5 Evaluate

Free to try, Paid


Reflection: In this activity, students are to visit flyers and blogs of peers and offer feedback. Students can ask questions regarding process or write additional questions related to the work of their peers. 

What if you were given the area or arc length and with a radius or degree? Could you work backwards? How  and give an example.

Extension: Check out this interactive graph about the cost of pizza as related to area. How might your pricing by degree or area look? Post it to your blog!