 3.2.30.a1 Definitions 0 Contents 3 Observing 3.2 Level 30 Curiosity Examples 3.2.30.a3 ## 3.2.30.a2 What Is The Same About All Circles?

Created 2007 5 Stavropol, Russia Last Revised 2007

 Investigation 1 Curiosity, 2 Purpose, 3 Materials, 4 Apparatus, 5 Procedure, 6 Observations, 7 Analysis, 8 Conclusions, 9 Discussion.

 Curiosity: 1 Top Next What Is The Same About All Circles?

Exposure Level:

At thirteen I was shooting hoops with a basketball in the driveway when I had a sudden realization. When I was close the hoop looked one size and when I moved away it was another size but something had to be the same. What was it? The same insight refers to the Basketball also. What is the same about all spheres? Figure 1 Circles and Lengths.

The circle is a special gift for children (or adults) to discover. If you haven't been violated by some "parental authority figure" into memorizing relationships in the circle take the time to discover now. Look at the circles and tell your self (speaking openly so you hear your own words) what relationships you can see. Is there any relationship between parts of the circle that a teacher guided your attention into discovering or per chance you discovered on your own?

 Purpose: 2 Previous Next The purpose of this experiment is to investigate the circle. This investigation is to explore and quantify the relationship between the distance across a circle and the distance around a circle.   We would like to discover a relationship where if we know the distance across a circle we could predict the distance around the circle. Figure 2 Circles.

 Materials: 3 Previous Next 1) a work table, 2) a compass to draw circles, 3) a pencil and paper, 4) a straight edge, 5) The "golden arm-length" or another standard measuring instrument, 6) a set of four logs segments with different diameters and  7) a string..

 Apparatus Diagram: 4 Previous Next   Figure 3 Cylinders and a measuring line.

 Procedure (observations): 5 Previous Next Using the measuring string, measure and record the diameter and circumference of each cylinder.

 Observations: 6 Previous Next Cylinder Diameter (A) Circumference (A) 1     2     3     4 0.26   0.52   0.86   1.15 0.816   1.633   2.700   3.611

Table 1 The measure of  4 cylinder's diameter and circumference.

 Analysis (observations): 7 Previous Next Cylinder Circumference (A)/Diameter (A) C/D 1     2     3     4 0.816/ 0.26   1.633/ 0.52   2.700/0.86   3.611/ 1.15 = = = = 3.138 3.140 3.139 3.140

(3.5 Table 2) The measure of  4 cylinder's diameter and circumference.

Discovery 1   C/D 3.14

Equation 1 C = D.

We say the circumference of a circle equals pi times the diameter.

We define the radius (r) of a circle as the the straight line distance from the center of the circle to the periphery (outer edge)

Equation 2 D = 2 r.

Using  Equation 2  (D = 2 r ) in Equation 1 (C = D) gives

Equation 3 C =  2 r.

 Conclusions: 8 Previous Next Given diameter D and circumference C.

Discovery 1   C/D 3.14

Equation 1 C = D.

We say the circumference of a circle equals pi times the diameter.

We define the radius (r) of a circle as the the straight line distance from the center of the circle to the periphery (outer edge)

Equation 2 D = 2 r.

Equation 3 C =  2 r.

 Discussion: 9 Previous Top Remember: (Discovery 1)  C/D 3.14 is a calculated abstraction! The reality is the measurable circumference and diameter.   