Pi and the Circle

Pi (\pi \approx 3.1416) is probably the most popular among all the irrational numbers.  It is seen in almost all fields of mathematics and it often appears in places that you least expect it to be (like in birthday party perhaps?).

Pi pie

\pi is the ratio of the circumference (C) of a circle and its diameter (d). That is, if we take a circle of any size, measure its circumference, measure its centimeter and divide them*, then

\displaystyle\frac{C}{d} = \pi.

This means that C = \pi d. Since the diameter of a circle is twice its radius (r), then it follows that C = \pi (2r) = 2 \pi r.

The area of a circle can be approximated by dividing it into equal sectors and rearranging it as shown in the first applet below. As we can see, as we increase the number if divisions (n) by moving the slider in the applet, the arrangement of the sectors is getting closer and closer to the shape of a rectangle.

[iframe http://mathandmultimedia.com/wp-content/uploads/2011/10/areaofcircle.html 600 500]

We can say that the “limit of the shape” of the rearranged sectors as d increases without bound is the shape of the rectangle.  Notice that the altitude is r, and its base is \pi r (Can you see why?).  The area of a rectangle is the product of its base and its altitude; therefore, the area (A) of the circle with radius r is expressed by the equation

A = \pi r(r) = \pi r^2.

Hence, the area of a circle with radius 1 equals \pi.

\pi can be approximated by inscribing a regular polygon in a circle with radius 1 as shown below.  By increasing the number of sides (n) — not to be confused the number of sectors — of the inscribed polygon, the polygon approximates the  circle.

[iframe http://mathandmultimedia.com/wp-content/uploads/2011/10/approximatingpi2.html 550 500]

Consequently, as n  increases, the area of the regular polygon gets closer and closer to \pi.  In fact, the limit of the area of the inscribed polygon as n increases without bound is equal to \pi.


First Applet courtesy of Linda Fahlberg

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