Area of Circles Measuring the surface in squares
"AREA is a measure of a surface using squares. The size of the square is determined by the linear units used to measure the length of the sides of the object."
We have a slight problem with cirles - they have no obvious sides.
Let's review what we do know about the parts of a circle. These parts include the Circumference which is the distance around the circle, The Diameter which is the distance across the circle through the center, and the Radius, the distance from the center of the circle to the edge.
| The first question we need to consider then, is what can we call a "side" of the circle? The circumference can't work - it is not a straight line. That leaves the diameter and the radius. If we used the diameter as a side, we would end up with a box around the circle - that doesn't help much. |
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| If we cannot use circumference, and diameter is too big, that only leaves the radius. Using the radius as the length of a side looks like something that might work - but the solution is not really obvious. |
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Here is one solution. (You might have a better one!) To start, lets use the radius to divide the circle into 16 equal parts. (The number 16 is not important - it is just easy to use.) Now our cicle would look like this.
Cut it apart and re-assemble it like this:
| Even with only 16 parts it is beginning to look like a figure from the other page called a parallelogram. This is the general idea - however 16 parts does not make a circle into a parallelogram. TO make the figure more accurate, i.e., to make it more like a real parallelogram, we need to cut each part in this pacture into smaller parts: (this picture shows what one of the "sixteenths" would look like cut into 4 smaller pieces. |
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As you can see, it is getting closer to having straight line sides. Using our imagination, we might be able to see that if the circle was cut into a million, or ten-million, or maybe more pieces and put back together this way, we would end up with a parallelogram with straight sides. Not only that, we know that the height of this shape is the same as the radius of the circle. What about the other side?
We are using the picture with the 16 pieces because it is easier to see and talk about then say, a million pieces would be. Remember, the circle was cut into 16 pieces and reassembled into a parallelogram. Notice that the bottome edge is made up of eight (8) pieces and the top edje is made of eight (8) pieces. In other words, the top edge is half of the pieces, the bottom edje is half of the pieces.
These edges are made from the outside or circumference of the circle. This means that the top side of the parallelogram is equal to half of the circumference of the circle (and so is the bottom!). Here we are going to need a little bit of help from math to find our number.
Going back to the information we know about circles: The diameter of the circle is twice as big as the radius, the radius is half of the diameter, and the circumference is equal to the diameter times the number named after the Greek letter 
(pi).
- d = 2 x r
- r = d ÷ 2
- C = x d
We have to make a substitution here. The formula for circumference says that it is equal to times the diameter. Since the length of base of the parallelogram is ½ of the circumference, we might want to make the problem a bit simpler. See if this picture helps:
This brings us to our final area formula for this lesson: