NAME______
For this activity, we are going to be using Geometer’s Sketchpad to construct various geometrical figures to discover important properties of circles and their proportions. This worksheet will give step by step instructions on how to construct these figures and manipulate them.
1)Open Geometer’s Sketchpad and open a new Sketch.
2)Construct a circle by selecting the 3rd button down on the right hand side and clicking and dragging a circle on the sketchpad.
Remember, when selecting a figure on the sketchpad, make sure you are selecting only the figures that you want to select. To deselect everything, click on the “selection arrow tool” and click on the white portion of the sketchpad. If at any time you want to undo your last operation, choose the undo option from the edit menu.
3)Click on the text button marked “A” and click the center of the circle, and then click the point on the circle. Your circle should now have two points labeled A and B.
4)Now construct another circle with center B. On circle B, label this point C.
5)Construct segments AB, BC, and AC by choosing the straightedge tool and clicking on both points to connect them.
6)Now measure the length of each of these segments by clicking on the segment and choosing Measure -> Length. Do this for all three segments.
7)Drag the points so that the circles change in size. Look at the relationship between the segments and circles. Note any discoveries you have about the segments and the circles:
8)Manipulate the circles so that they look about the same size. What do you notice about segments AB and BC?
9)How are these segments related to the circles?
10)Move point A so that is on circle B, and move point C so that it is on circle A and B. Make sure point A is still on circle B. What particular figure did this form? Be specific.
11) Click on both circles and choose Measure -> Circumferences
12) Click on both circles and choose Measure -> Areas
13) Manipulate the segments so that AB + AC = BC (Move point A so that it is in the middle of B and C). What do you notice about the circumference of Circle B in relation to the Circle A?
What do you notice about the area of Circle B in relation to Circle A?
14) Manipulate the segments so that AB – AC = BC (Move point B so that it is on top of point C). What happened to circle B?
Why did this happen? (Think of the radii)
15) Manipulate the circles so that the Circumference of Circle B is 3 times as big as Circle A. What is the relationship between the radius of B and the radius of A? What is the relationship between the area of B and the area of A?
16) Based on this exercise, how would you best explain the relationship between radii, circumference, and area?
From this activity, we have discovered the relationship between the radius, circumference, and area of a circle. Later on, we will see that for every circle, there is a special number (or constant) in relation to the radii that allows us to calculate and generalize the circumference and area of any given circle.
ANSWER KEY
7) The segments are also radii of Circle A and Circle B. It seems that the size of the circles are related to the segments that are radii. As one radii gets bigger, the circle gets bigger and vice versa.
8) They are about the same size
9) They are the radii of the circles (AB is radius of circle A, BC is radius of circle B)
10) An equilateral triangle
13) The circumference of B is twice circle A.
The area of circle B is four times circle A.
14) Circle B disappeared
The radius of circle B has a length of 0 (BC has a length of 0)
15) The radius of B is 3 times as big as circle A. The circumference is also three times as big. The area of circle B is now 9 times as big as circle A
16) The circumference and area of a circle depend on it’s radius. Comparing two circles, if the radius of one circle is twice the other, the circumference will be twice the other. The area will be how many times the radius is bigger than the other circle squared.
In this activity, I want students to discover the relationship between radius, circumference, and area. They do this by comparing two different circles and manipulating their radii. I am not trying to get them to find the formula, but merely see that the relationships depend on the radii, and that the relationship for circumference and area are a bit different. This will help them better understand the formula for circumference and area when it is introduced to them. This can also be an activity used as they learn the formula so they could see it for themselves. The question with the equilateral triangle seems to be erroneous, but I thought it could be another cool discovery between radii and shapes for the students to look at.