USING THE GEOMETER'S SKETCHPAD TO EXPLORE CIRCLES
As a review, show that you can:
- create a segment, ray and line;
- show or hide the labels of points;
- construct the interior of the figure, color it green, and measure the area.
To illustrate how the Sketchpad can be used to explore geometric relationships in circles, follow the sequence of steps outlined below:
- Begin with a new sketch. In Display >> Preferences, turn Autoshow Labels ON for points. Make sure the unit for Distance is inches, with Precision to the hundreth; Make sure the unit for Angle Measure is degrees, with the precision of units.
- Construct a circle. The center will be labeled A and the control point will be point B. Construct the radius, segment AB. Make this segment thick and red.
- Place an additional point, C, on the circle. To construct a diameter, use the line tool to construct a line that goes from point A and passes through point C. Shift+select the line and the circle, and construct their point of intersection, point D. Hide the line, and connect points C and D with a segment. Make this segment thick and blue.
- Measure the lengths of both the radius and the diameter. Also measure the circumference.
- Use the calculator to calculate the ratio of (Circumference ÷ Radius).
- What value do you get? Did you get the same value as others around you? What happens to the value as you grab-&-drag the center and control point to change the size of the circle?
- How do you explain what's going on here? How does this relate to the formula for the circumference of a circle?
- Select the circle; Construct >> Circle Interior. Color it yellow. Measure the area inside the circle.
- What is the formula for the area of a circle? How could we use the calculator and the measures that are showing to calculate the area of the yellow circle?
- Can you make a triangle that has the same perimeter as the circumference of the circle? Which is greater, the area of the circle or the area of the triangle?
- Can you make a quadrilateral who's perimeter is the same as the circumference of the circle? Which is greater, the area of the circle or the area of the quadrilateral?
What if we tried a pentagon, hexagon, 100-sided polygon, etc.
What general conclusion do you think this leads to? For example, try to describe how to enclose the greatest area you can within a given amount of fencing.