Archimedes on the Area of a Circle
Use the checkboxes to inscribe or circumscribe regular polygons in the circle. The area of the polygon is displayed numerically and plotted on the "Area" graph. Use the slider to change the number of vertices of the polygons. The "Area" graph represents the areas as numbers, plotting them as points on a line.
Note that the circumscribed polygons are bounded below by the circle. The inscribed polygons are bounded above by the circle. The relationship of the circle to the polygons is as their "least upper bound" (inscribed) and "greatest lower bound" (circumscribed) respectively. In both cases the circle is the "limit" or "bound" of the polygons. The "Area" graph represents the areas as numbers. In this way, a 2-dimensional concept (area of polygon) can be described as a 1-dimensional number. The area of a circle with radius 1 is π, and the points plotted on the line bracket and approximate π. Do you see how the relationship between the points plotted on the "Area" graph and π is exactly the same as that between the polygons and circle? The point corresponding to π on the graph does not form a visible "limit" or "bound" to the plotted points, but it is their "limit," just as the circle is to the polygons. A similar use of the word "limit" will appear in the definition of continuity of a function.