Finding the area of a segment.

Today is the last day of instruction as we only have one more class before the final exam.

We started by checking results from the homework problems. Students seemed to do well with these. I dove into the idea of finding the area of a segment within a circle as this involves review material that students need for the final.

I revisited the overlapping circle scenario and told students to focus on the sectors that overlap. In essence, although the circles are overlapping, what we really need to be concerned with are the sectors that overlap. I colored in the segments that are formed and asked students to consider what could be done to find the area of the segments.

As I walked around, many groups were thinking about the triangle that gets formed. I was pleased that they were considering how to make use of the triangle. One group worked out the idea of finding the area of the sector and then subtracting the area of the triangle. (see image below)

I asked what the area of the triangle would be if the central angle formed by the sector was 50^o and the radius was 6 units?

As students looked at this situation, I asked the class what they needed to know about the triangle to calculate the area. Most of the students recognized they needed the base and the height. The base of the triangle is the chord forming the segment. The height is the perpendicular bisector from the circle's center to the chord.

With these pieces in place, students started to realize they would need to use trigonometric ratios to find the missing lengths. This took some time but students started to see that the triangle's height was found using the cos(25^o) and the base could be found using sin(25^o).

I provided two more problems for additional practice, using 60^o and 40^o with radii of 3 and 5, respectively.

With that, we used the remainder of the class to take the circle test. I had students work 20 minutes on their own. At this point, I had students circle the problems they had completed on their own. I then gave the class an additional 20 minutes to work in their groups. They were not to give each other answers. What I asked them to do was to ask each other questions. For example, in the equation of a circle, if they couldn't remember whether the constant value represented the radius or the square of the radius, they could ask their partners that question for clarification. I was hoping to focus their collaboration and growing their learning.

Next class will be looking at more review problems and then we'll take the final two days after that.

I'll be outlining where I was taking the rest of the circle unit and how I was going to tie it into function transformations in my next post.