The Greek mathematician Archimedes approximated pi by inscribing and circumscribing polygons about a circle and calculating their perimeters. Similarly, the value of pi can be approximated by calculating the areas of inscribed and circumscribed polygons. This activity allows for the investigation and comparison of both methods.
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In this applet, polygons are inscribed and circumscribed around circles. The area of the polygons around a circle of radius 1 are calculated on the left, and the perimeter of the polygons around a circle of diameter 1 are calculated on the right.
Change the value of n to increase or decrease the number of sides in the polygons, and notice how the calculations of the areas and perimeters begin to approximate pi.
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The inscribed polygon is blue, and the circumscribed polygon is red. Not surprisingly, the blue polygon gives an underestimate, and the red polygon gives an overestimate. But by how much?
- Choose an arbitrary value of n. For the area model, which is closer to the actual value of π, the estimate given by the blue or red polygon? At what point between the blue and red estimates does the actual value of π occur?
- For the perimeter model, which is closer to the actual value of π? At what point between the blue and red estimates does the actual value of π occur?
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