Gr 06 Deriving a Formula for the Area of a Circle

Yesterday we focused on the names of parts of the area of a circle. Today we are focusing on one of those areas: the quadrant.

1. Ask students to title and label their math notebook page. Then they construct a 4-cm radius circle around point A.
2. Students then construct a quadrant of their circle using two radii that are 90-deg apart.
3. Verify with students that each radius is 4-cm long, then ask them to build a square out of that radius. They simply add lines perpendicular to the ones they have. But then verify with them the length of those new sides.
4. Define this shape with them as a radius-square because it is built on the radius of a circles.
5. Ask students, "how would I find the area of this square?" They'll be quick to offer up that it's 16 sq units. But ask how they did it and they'll say 4x4. Write that down. Then ask them what parts of the circle these two 4's are. The kids will say they are the radius.
6. Then replace those numbers with r's to represent radius and explain to them (I know, this is the short version) that in math we can write A=rxr as A=r².
   
   
7. Now comes the fun part. Ask them how many radius-squares they think they can fit inside a circle! I prepared a circle and some radius squares for them to cut out. They need to use up a radius square completely before moving on to the next one. I tell them to use the blue one only if the have to.
   f
   f
8. When they are done, the should have an answer: that they can easily fit three of them inside a circle plus a little bit more. And that will make some students say, "PI!" And they are right!
   
   
9. So if you wanted to, you could have them use up some of that blue square to fill in any spaces, but it's really hard to do. It is often good enough to stop here and formalize then the relationship.
   
10. And that relationship is that it takes about three radius-squares to fill a circle. More precisely, it takes pi radius-squares to do it. Or
    A= π x r².