In the last post, we established the relationship between the diameter and the circumference of a circle, i.e. the circumference of a circle is always π times as long as the diameter of the circle. Unlike the area formula for polygons we have looked at previously, to establish the formula for the area of circles, we need to know something about the circumference as it will become clear a little later.
But, before we get to the formula, let's investigate the area of a circle a bit more intuitively. The picture below shows a circle with an inscribed square (black) and a circumscribed square (red):
If you compare the area of these two squares to the square that has the radius of the circle as a side (shaded), you see that the inscribed circle has the area twice of the shaded square, and the area of the circumscribed square is 4 times of the shaded square.
It should be clear that the area of the circle is greater than the area of the inscribed square but less than the area of the circumscribed square. Since the area of the shaded square is (radius)^2 (I can't figure out how to make "2" into a superscript...), we can say the following about the area of circle:
[Note: the exponentiation notation isn't discussed in elementary schools, so it is probably better to write it (radius x radius).] If you draw a circle on a grid paper (let's say with the radius of 10 cm), you can refine this approximation even further. The area of circle turns out to be a little more than 3 times of (radius)^2.
In everyday situation, knowing that the area of a circle is a little more than 3 times of (radius)^2 may be good enough. However, sometimes you may want to know more exact value. So, how can we derive the formula for the area of circles? How can we change a circle into a familiar shape? There are a few different possible routes, but I will discuss the most common (I think) approach.
Suppose you have some pizzas left over. Your refrigerator is too full to put the whole box in. What would you do? One thing you might do is to re-arrange the slices like the picture below shows:
When we rearrange, we know that we still have just as much pizza as we did before. But the resulting shape looks more like familiar shapes we have seen before, in this case, a trapezoid. So, we try to use this idea as we derive the formula for the area of a circle.
Suppose we subdivide a circle into 6 equal sectors, then re-arrange those sectors. It will look like this:
If we cut the same circle into 8 sectors then re-arrange the sectors, it will look like this:
If we keep increasing the number of sectors, each sector will get thinner and thinner. Here are the pictures showing what happens when the same circle is cut into 12 sectors and then 24 sectors:
As the number of sectors increases, the shape we get after we re-arrange them will get closer and closer to a rectangle:
Because the re-arranging of sectors do not change the area, the area of this rectangle is equal to the area of the original circle. So, we just need to calculate the area of this rectangle. The area of rectangle can be calculated by multiplying its length and width. By paying attention to where these parts came from the original circle, we see that the length (vertical dimension in this picture) is equal to the radius of the circle, while the width is a half of the circumference of the original circle. Therefore,
But, we already know that,
By substituting the second one in our formula for the area of a circle (because we only want a half of circumference), we end up with the formula for the area of a circle,
The process of changing the given shape into a familiar one used here is a bit different from what we did with polygons. In fact, it will probably be very difficult for children to actually cut circles and re-arrange the sectors - it can be done by providing circles with the sectors already drawn. However, even when we provide pre-drawn sectors, realistically, we can cut the circle into 12 or so sectors at most. At that point, the result may not look like a rectangle - it may look much more like a parallelogram. Since we already know the formula for the area of a parallelogram, we can also use that idea, too.
Deriving the area formula for circles is definitely more complicated than deriving other formulas. Unlike other formulas, there may have to be more "demonstration" than actual hands-on activities. Some may question whether or not it is an appropriate learning goal for Grade 5. However, it is a Grade 5 standard in the GPS, and we need to think about how we can make the formula more meaningful to students. What we discussed here is just one approach to deriving the formula. I encourage readers to investigate other ways of approaching this topic.