Computing Pi

  • Computing Pi

    Grade: 6th to 8th, High School

    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.

    Activity

    Instructions

    • In this applet, polygons are inscribed and circumscribed around circles of radius and diameter 1.
    • Underneath the diagrams, the area and perimeter (labeled "Circumference") of the polygons are calculated on a number line. Note that the location of pi is also shown for reference.
    • Use the blue slider to change the value of n, the number of sides in the polygons, and notice how the calculations of the areas and perimeters begin to approximate pi.

    Exploration

    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?

    Objectives and Standards

    NCTM Standards and Expectations
    • Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
    • 6-8
    • High School (9-12)
    • Geometry