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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.
In the spirit of Archimedes’ method of approximating pi, students inscribe and circumscribe regular polygons in and around the unit circle, which is known to have an area of
. Students then consider the area of the polygons, using either an applet or a graphing calculator. The area of then‑gon will approach
as n increases. Similarly, students consider the perimeter of inscribed and circumscribed regular polygons in and around a circle with unit diameter. This exploration also leads to an approximation of
. Taken together, the two methods provide a compelling investigation of a method for generating the never‑ending digits of
Use Cavalieri's Principle to Compare Aquarium Volumes
This task presents a context that leads students toward discovery of the formula for calculating the volume of a sphere. Students who are given this task must be familiar with the formula for the volume of a cylinder, the formula for the volume of a cone, and Cavalieri’s principle.
This task involves several different types of geometric knowledge and problem-solving: finding areas of sectors of circles (G-C.5), using trigonometric ratios to solve right triangles (G-SRT.8), and decomposing a complicated figure involving multiple circular arcs into parts whose areas can be found (MP.7). It also engages students in some of the steps of the modeling cycle.
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