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Course: NASA>Unit 2

Lesson 3: Orbital mechanics

Discovering pi

Early models of the universe were based on the assumption that circles were perfect models for the orbit of planets.

Anatomy of a circle

Circles are symmetric in every direction, and all points on a circle have the same distance from its centre – this distance is called the radius. The distance around the perimeter of a circle is called the circumference.
The most fascinating feature of circles is based on the ratio between the circumference and diameter. What do these circles have in common?
The ratio between the circumference and the diameter is the same for circles of any size.
It is a mysterious number which is represented using the Greek letter  π, but what does π (pi) equal? To find out, look for a circle and divide the circumference by the diameter. You will find it is always around 3. Old Babylonian mathematics contained some estimates for  π. Tablets show they assumed π  = 25/8 which is 3.125.
Can we do better?

How to estimate π

If you draw a polygon inside a circle, you would expect the perimeter of the polygon to be approximately equal to the circumference of the circle. This idea was developed by Archimedes (287 BC –  212 BC) who used a 96 side polygon to improve the estimate for pi. To do so he compared the perimeter of the outer polygon and inner polygon. The circumference of the circle must be between these two values.
Below is an intereactive illustration. Use the slider to change the number of sides and notice how this affects the estimate of π
As you increase the number of sides your estimate becomes more and more accurate. This leads to the following estimate for π (3.1408 < π < 3.1429) or approx 3.14
Using more sides will result in a more precise estimation of π.
An even better estimation which is handy to memorize:

Want to join the conversation?

• why did they make pi? but it is used in every day life.
• Pi allows us to do calculations such as finding the diameter of a tree by measuring its circumference, or finding the surface area of the earth, by doing nothing more than multiplying or dividing by pi. Trigonometry uses it constantly; it turns up in probability, series, and physics, among other places.
• Is there a Taylor Series for Pi? If there is, what is it?
• As Pi is not a definitive value should perhaps another or more factors , suitable, involving a circle if there are any at all, be involved in determining a useable constant -- something other than pi that would function similar to pi ? Havent given this query much thought, just sort of off the cuff. An awful lot of time seems to be spent trying to end this likely infinitive equation.
• Pi is an acceptable value to use in calculations for circles because it has such a large number of significant digits.

The usefulness of it depends on the size of the circle one is measuring. Obviously, the larger the size of the circle the more significant digits you will need if you want accurate measurements. Because Pi has such a large number of significant digits it's unlikely that we'll ever find any circles for which it will not be useful.
• Why did Archimedes use 96 sides for his polygon instead of 100?
• Greece had a base-12 counting system those days.
• Is pi a constant without dimensions eg speed or time.
• Pi is a constant. (I think? Unless there's another, non-algebra, way to define it.) A constant is a value in an equation that doesn't change, like e or i or 3. As a number, it is a constant. Speed is not a dimension, though, so I don't know what you mean by this.
• Why is pi pi? Why did the greeks name pi pi?
(1 vote)
• Pi (the number) is represented by the Greek (lower case) letter `π`, and is derived from the first letter of the Greek word for circumference. People just usually spell out π as `pi` when they don't write out the symbol `π`.
• What is circumference, diameter, and radius
• The circumference is the length of the outer edge of the circle. A diameter is how long the line from one side of the circle, through the center, and touching the other side is. The radius is the length between the center and the side of the circle. Credits to Preston on the comment
(1 vote)
• pi is also denoted as rational number 22/7 . how do u explain that
• That is just an approximation. 22/7 is equal to 3.142857..., while pi is equal to 3.141592... See the difference?
• i know Pi is infinite, but how far do the numbers go? i know there are more than one type of infinite, so which one is Pi?
• Infinite is a concept about something that's endless or neverending.
That said, `pi` is an irrational number, which means its decimal expansion is infinite and doesn't have an endlessly repeating pattern. As of March 14, 2024, about 105 trillion digits of `pi` have been calculated.
`pi` of course isn't infinite in value, since `4 > pi`.

Bonus: Compare an approximation of `pi`, `22/7`, to `pi`.
- `22/7 = 3.142857142857142857...`
- .__`pi = 3.141592653589793238...`

Hope this helped!
(1 vote)
• Isn't The Whole Pi Number so long it is impossible to write it completely?