What’s the Area of a Circle?

It has been a while since I have had time to post. This semester is getting crazier and crazier as it nears the end.

But anyway, we have been working on a measurement unit the past couple of weeks with our 7th graders. We started with finding the area of the different types of polygons; rectangles, parallelograms, triangles, and trapezoids. We spent the first part of each class deriving the area of the polygons by transforming them into shapes that we already knew. Students started out by cutting out the shapes on graph paper and then manipulating them to turn them in shapes they already know the area formula for. These activities came from my CT, check our her blog post on it here.

I also got a great idea for deriving the area of a circle from a fellow math TA in the program. We started with a circle that that was divided into 12 equal sections. The last section was also divided into two equal parts. When I did this I had labeled those sections as 12 and 13, but after thinking about how the lesson went and talking with my CT I think it is easier to label the two smaller sections as 12a and 12b.

Circle Uncut labeled 

Here’s a copy of the circle that we used. Circle Document 

I then cut the circle into these sections and lined them up so that they formed a shape that closely resembled a rectangle, although it wasn’t perfect because it was only cut up into 12 pieces.

Circle Proof 

We then talked about what the height would be for this rectangular shaped we created. I was impressed at how quickly they identified it as the radius of the circle. We then talked about what the base of the shape would be. This wasn’t as smooth as the height but they still got it. We talked about how many of the arcs made up each base, which they found was 6 and when we compared it back to the uncut circle and found that the base would be about half the circumference of the circle. We used what we knew about the area of a rectangle and found that the area would be:

A = r*.5C.

A= r*.5(2πr)

A= πr^2.

I was really pleased with how this activity worked out. This was something that I had never seen before and I wasn’t sure how it would go since the rectangle made wasn’t as clear cut as the previous shapes we had used. It is something that I most definitely plan on using in the future.

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