Arc Length and Area of a Sector (Lesson 9.9)
To complete today’s lesson, students will need to know the formula for the circumference and area of a circle. You could start the class with a warm up about that to be sure all students have the formulas. From there, students can work through the whole activity before debriefing as a class. The activity uses degrees only. For groups that finish early, they can work on the challenge of writing formulas using radians instead of degrees.
The circumference of a circle is found using C = 2πr
The area of a circle is found using A = πr^2.
What’s another word for the whole distance around a circle?
How could we find that?
What fraction of the whole circle is the angle?
So what fraction is the pizza slice of the whole circle?
To write a formula, let’s look at your work.
Where did the 7 come from? Where did the 62 come from?
With our formalizations today, we want to make sure we don’t get too formal. I know that sounds confusing, but what I mean is once you give students a formula, they often rely too heavily on it. Instead of memorizing the formulas, we really just want them thinking proportionally about what fraction of the circle they have and how that relates to arc length or area.
Step 1: Find circumference or area of the whole circle.
Step 2: Find the fraction of the whole circle the sector is.
Step 3: Multiply the fraction by the circumference or area of the whole circle.
While a formula may be more efficient in the short term, developing this proportional thinking is much more reliable in the long term. So while you can discuss the formulas in the challenge question, we suggest leaving it out of the QuickNotes.