Project

A major application of pi is that it is essential in finding areas and volumes. The area of a circle is given by the equation \(A=\pi r^2\). We will explore why this is. From the drawing below we see that a circle can be partitioned into many isosceles triangle-shaped sections. Since the sides that are congruent go from the center point of the circle to the outer edge, by the definition of a circle, each side of the triangle has a side length of r. As the number of triangles increases, the triangles become smaller and they can be arranged into a parallelogram. The height of this parallelogram is the length of one of the sides of the triangle, which is r. The length of this parallelogram is one half of the circumference of the circle. The circumference of the circle is \(2\pi r\), thus one half of the circumference is \(\pi r\). Thus the area of this parallelogram, which is the area of the circle rearranged, is \(\pi r\cdot r\), or \(\pi r^2\) (Cuemath).

Since pi is essential in finding the area of a circle, it is also fundamental in finding the surface area of shapes containing circles, such as cylinders and cones. It is also essential in finding the volumes of these shapes. The formulas for all of these surface areas and volumes include pi (Mometrix). While being able to calculate the surface area and volumes of three dimensional shapes is very useful for your geometry test, it begs the question- when is this applicable in real life? The answer is that it is applicable more often than you might initially think. Surface area and volume are used in everyday life: how much wrapping paper it takes to wrap a present, how much paint to use, how much liquid is needed to fill a container. There are also many practical applications for these concepts in construction and engineering, and even in rocket ships. In fact, NASA has an entire webpage dedicated to how pi is used in their work. Pi is used to determine the volume, surface area, and density of planets. Once scientists have determined the density of planets, they are able to determine which materials the planet is made of. Scientists also use pi to study craters. They work to determine how circular a given crater is using pi and the crater's perimeter and area. From these calculations, "planetary geologists can reveal clues about how the crater was formed and the surface that was impacted." Pi is essential in finding surface area and volumes and therefore understanding the universe around us (NASA).

Another application of pi is converting radians to degrees. Most high school students learn how to convert radians to degrees and vice versa by using the conversion factor of \(180= \pi\) radians. However, what most high school students do not learn is why that is and why radians are useful. "A radian is defined as an arc that has the same measure as the radius of a circle" (Mometrix). We know from the definition of pi that \(\pi=\frac{c}{d}\), which can be rearranged to show that the circumference of a circle is given by \(C=\pi d\). We also know that \(d=2 r\) where r is the radius. So by substitution we can find that \(C=2\pi r\) and so the measure of the entire circle is \(2\pi \) radians. The measure of an entire circle is also defined to be 360 degrees, and so \(360=2\pi \) radians. It is from this relationship that we get the conversion factor most commonly used: \(180=\pi\) radians. Knowing where this relationship comes from makes the conversion factor much more intuitive.

Radians are significant because they have meaning. Degree is a measure that is arbitrarily assigned- different societies have chosen \(400^\circ\) as the measure of a circle or \(200^\circ\) as the measure of a circle. The only meaning that a degree has is an amount that is arbitrarily assigned to it. In contrast, radians have meaning inherent in their definition and structure. A circle will always be exactly \(2\pi \) radians. In addition, if the circle is a unit circle then the length of the arc subtended by the central angle of the circle is the radian measure of the angle. Radians are useful in calculus and in trigonometry. Graphs of sine and cosine are much easier to make sense of if they are done using radians rather than degrees. If degrees are used for these graphs, the horizontal axis has to be stretched to be able to see the typical curve for a sine or cosine graph, whereas a typical graph with both the horizontal and vertical axes scaled the same functions well for radians. Radians are more useful for calculus as well because their derivatives work out much nicer, whereas if degrees are used there will always be a multiplier of 180 complicating the equations (McMullin).

Another application of pi is closely connected to radians as well as sine functions- waves. Waves are described using radians, and thus pi. The function for the period of a wave is T= Period = \(\frac{2\pi}{k}\) where k is the angular frequency. This equation is one that most physics students memorize. However, what many students may not realize is that this equation is derived from the relationship of pi and radians. Since one circle is complete in \(2\pi\) radians, it is also true that a full cycle of the wave is completed at \(2\pi\) radians, and thus a sine or cosine wave has a period of \(2\pi\) radians. It is possible to parameterize and understand waves because of our understanding of pi.