In order to teach finding the area of a circle, I used and modified a couple of activities out of the “Hands on Math” book. Look under the “Resources” tab for a picture and review of this book. It is an excellent resource. Before I teach finding the area of a circle, I first teach my lesson on finding the circumference of a circle. Doing this, they are already familiar with pi and know that it is “three plus a little bit more.” The first activity we do is called “Circle Cover-Up.” For this activity, the students are groups of two but they each do the activity themselves. The reason for pairing them is so that they can watch the other person to make sure they are doing the activity correctly. In the “Hands on Math” book, there are black-line masters of the pages I use. You print the pages out in two different bright colors so that they stand out. Below is a picture of the two pages we use. One is a centimeter grid and the other is two different sized circles with a radius drawn and tick marks to show the measurement of the radius.
I tell the students to cut out squares that have the same side length as the length of the radius. For example, if the radius is five units, then you would cut out a square that is five units long and five units wide. I usually go ahead and tell them to cut out four of those five by five squares out. They are actually suppose to cut each of the larger squares into the smaller squares and place each individual square on the circle. I found it much easier and less time consuming to have them cut the larger squares into strips or as big of pieces as they can. They are to glue these on the circle and make note of how many larger squares it takes to fit on the circle. This is kind of difficult and the students won’t all get the same answer. Most usually only fit three on the circle. This is fine because I look through the class and find the best example to show the students that they should have been able to get three larger squares plus a little bit of the fourth square.
The students are to do this with the second circle on the page. That circle has a radius of four, so they are to cut out squares that are four by four. Again the students should use three squares plus a little bit of the fourth one. I ask them if they remember that from learning circumference. By now the students have already figured it out that pi has something to do with it. I draw the following diagram on the board using the measurements from the first circle.
I explain to them that the meaning of the area of a circle is how many unit squares will actually fit onto that surface. I then draw an example of the squares we just did on that same circle. I know, I know, I am no artist, but the kids get the point of my drawings.
I ask them how many small squares are in the one larger square on that circle. They say twenty-five and I make sure they understand they easily find this by taking the side times the side. I question them and make sure they understand that the side length is also the same as the radius. I then draw that same square on the other three sections of the circle. I ask them if we actually was able to fit all four of those large squares on the circle. No, then how many? Three plus a little bit more. What do we call that three plus a little bit more? Pi. Very good. So if there are twenty-five in one square then there would be twenty-five in each of the other squares too right? How many big squares were we able to fit on? Pi. How did we calculate how many small squares were in the big square? Side length times the side length. How did we determine the side length? The radius. So you could also say the radius times the radius. So if there are twenty-five small squares in this large square, how many are there in another larg square? Twenty-five. How many in the other large square? Twenty-five. So how many altogether in those three large squares? Three times twent-five which is seventy-five. Did we use all four squares? No. Three plus a little bit more. Sooooooo…… I could find out how many little squares there are on the circle if I took the radius times the radius times three plus a little bit more (pi). We also call radius time the radius, radius squared (sorry, I can’t make the little squared number so I’m just going to have to write it out.). So we could make our formula…… A = r^2 * pi. Wow, that was quite a dialogue I just had with myself. I hope it didn’t confuse you. Maybe reading a few times will help you. It’s much easier when I’m teaching it to actually human beings who respond back to me. I hope you got the main point of it and are able to use this information in your classes.
We work several examples using different measurements for the radius. For the first several, I draw a circle and the squares in them until the students catch on and are able to understand it without the pictures.
The activity we play to review finding the area of a circle is the same game I used when finding the circumference of a circle only making them find the area instead of the circumference. This game consists of the following modified paper with two spinners on it, a large paper clip, and a pencil.
They spin the paper clip on the top spinner to get what dimension they have. They spin the paper clip on the bottom spinner to get the measurement and find the area using that information. I have found that doing this activity and the activity with circumference, the students actually understand the meaning and reasoning behind what those two measurements really are. When you combine area and circumference on the same worksheet is when most students become confused because they forget which formula goes with which measurement. I am able to remind them that with circumference, we had the string and wrapped it around the circle and then saw how many times the string could go all the way across the circle which is the diameter. With area, we cut up the squares to put on the circle and we needed the radius to find out how many squares were actually on the larger square. Hopefully this gives you some insight on a good way of teaching area. This activity is the best I’ve found so far.