FINDING p - UMass
FINDING ( with Circumscribed Regular Polygons
Objective: Students will understand a method for approximating п by finding the perimeter of a circumscribed regular polygon. This will help students understand what the number п is, improve their number sense, and give a use for the circumference of a polygon using its radius.
Assumed Knowledge: Students should know what the radius, apothem, central angle, and circumference of a circumscribed polygon is. Also, they should know how to find missing measurements of these items. Students are also assumed to know the formula of the circumference of a circle.
Why Use J Software: Students saw that finding the circumference of a polygon given its radius and the number of sides can be a tedious process. We will do this process many times for large numbers. J will allow us to eliminate mistakes and concentrate on the results of our experiment without getting bogged down in complicated trigonometry equations. Also, J software is a free download for educational use from .
The Lesson:
Remind the students of the terminology and process in finding the circumference of a circumscribed polygon.
[pic]
Remember we are actually using 1/2 the true central angle. That is why we use 180 / n and not 360 to find the central angle we will be using. In addition we will be looking at polygons with radius 1 to make life easy. After all, п is the same for circles of any radius. So 1/2 side = sin (central angle)
Note: All use of NB. in J means commentary and you do not need to enter it in your code. It is just to help you understand what is being done in J.
Let's use J software to compute the perimeter for us. Type in the following:
Load 'trig' NB. Allows us to use trig.
N =: ] NB. N for number of sides.
CentralAngle =: 180"_ % N NB. This gets the central angle for any amount of sides we want. Try it.
Central Angle 4
Good, now we have a way to get the central angle. Let's use it to find half the side of the polygon.
HalfSide =: sind @ CentralAngle NB. Find sin after finding the central angle.
Side =: Halfside * 2: NB. Get the Side Length
Perimeter =: Side * N NB. Multiply the side length by the number of sides. Try using this.
Perimeter 4
Perimeter 6
Perimeter 10
Now, we have a way to get the perimeter of any polygon of radius one. Let's look at how this relates to the circumference of a cimcumscribing circle.
[pic]
This is a rough sketch of what is happening. The more sides our polygon has the closer it will look like a circle. So, if we pick a large enough number of sides we can approximate the circumference of a circle. If we can estimate the circumference we should be able to approximate п.
Remember the formula for circumference is:
[pic] So, п can be found as [pic]. Bur we are using a radius of 1. So п = C/2.
This is good because we can then approximate pi by using perimeter instead of C. So, in J we can write:
Pi =: Perimeter % 2:
Pi 4 NB. this is the approximation
2.82843 using a square. But, Pi is about 3.1415. Let's try Polygons with more sides.
Pi 10
3.09017 NB. Better, but not good.
Now, have the students find the smallest number of sides to get an approximation good to 4 places past the decimal point. That is Pi = 3.1415. I get Pi 231. As you can see just 4 places of accuracy takes a lot of tries. The Greeks did this by drawing in the sand and using trig tables since paper was too expensive to use for experimentation. You can imagine how difficult this would have been. The use of computers has allowed for much better approximations of pi than would have ever been imaginable.
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