G.GMD.1 STUDENT NOTES WS #6 – geometrycommoncore 2
THE CIRCLE
The circle definite poses a different type of problem when trying to derive the area of it. It obviously needs to be approached differently from the polygons that we have already discussed because of its shape. We are going to use an informal limit argument to help us determine its area formula. Limits in general are an upper level mathematical idea but when presented in a visual way they can be understood at earlier levels.
CIRCLE – DISSECTION (LIMIT ARGUMENT)
Once again we are going to take the original shape and cut it up into smaller pieces to form a shape that we already know a lot about.
The first dissection cuts are 90° sectors. When we put the four pieces together to form a new shape we get something that isn’t very recognizable. / The second dissection cuts are 45° sectors. When we put the eight pieces together it seems to form a more recognizable shape… maybe a parallelogram but the sides are ‘bumpy’.The third dissection cuts are 30° sectors. This definitely is making the shape look a lot more like a parallelogram. / Finally, our fourth attempt we use 15° sectors. This makes the new shape more and more parallelogram like… that is the intent of the limit argument.
If we continued this process down to an angle size of 1° or even 0.0001° the new shape would approach the parallelogram shape. This is the limit argument. As we decrease the angle size of the sector the area begins to approach the shape of a parallelogram. Now lets look at the dimensions of that parallelogram. The height approaches the radius, r, of the circle. Notice how r is NOT the height in the second example but by the fourth example r looks to be very close to the height. Now we need to look at the base value. The base value is half of the circumference of the circle, the circumferene of the circle is half divided into two halves. If we put this together we get Area = ½ (2pr)(r) = pr2 using this informal limit argument.
/ Acircle = pr2
/
AREA OF A CIRCLE SECTOR
A sector of a circle is a part of the circle formed by two radii and the circumference. It looks like a pie piece. Often we are asked to calculate circle sectors whether it be half of a circle, a semicircle, or a quarter of a circle. These divisions are easy to visualize and calculate. If you want to calculate the area of a semicircle you divide the area by two or in the case of a quarter you divide by four. It becomes more difficult if the sector has a 49° central angle…. What do we divide by now?
Think of a circle as a percentage of the circles total area, so all we do it create a percentage based on the central angle to the total angle value, 360°. This calculates a circle sector. / AREACIRCLE SECTOR =
Find the AREA of each circle or circle sector or composite shape.
______(E) / ______(E) / ______(E)