1170 words - 5 pages

Prove the Circumference Formula

Introduction:

Archimedes is credited to be the creator of the circumference formula, but more importantly to find the first theoretical calculation of Pi. Pi is an irrational number and the digits are continuous and never come to an end. Archimedes knew he had only found an approximation of pi, he found that pi is between 3 1/7 and 3 10/71. “Pi is a name given to the ratio of the circumference of a circle to the diameter. That means, for any circle, you can divide the circumference (the distance around the circle) by the diameter and always get exactly the same number. It doesn't matter how big or small the circle is, Pi remains the same. Pi is often written using the symbol ￼ and is pronounced "pie", just like the dessert”(math.com).

The circumference formula we use to this day is:

￼

Pi is abbreviated to 3.14 when being written, but calculators use a much more accurate version of the number.

I chose to investigate this topic because the origin of formulas interests me. Somehow these letters are created and it works under any conditions and never seems to fail. Prior to this investigation I have learned what the circumference formula is and how to apply it.

Statement of Purpose:

The main purpose of this investigation is to prove the circumference formula to be correct. Through this investigation I will use different processes of math to prove this formula correct. This will show that the formula holds true in multiple settings.

Plan of Investigation:

I will derive the formula and work it in multiple ways to prove that the formula we have used for the past centuries to be correct. I will also look at the history of pi.

Math:

￼

The above picture shows all the parts you need to know for this exploration. ￼

Pi is the Circumference (c) of any circle divided by its diameter (d)

￼ for any circle

With the above formula you want to rearrange it to solve for c the circumference

￼

The diameter of a circle is twice the radius of a circle so the (r) replaces (d) and that is how you get the circumference formula of:

￼

The below table are different circles diameters being used to show that no matter how big or small it is the circumference is always able to be found.

Diameter Radius Circumference

￼

8”

8”/2= 4”

2π(4”)= 25.13274123”

15” 15”/2= 7.5” 2π(7.5“)= 47.1238898”

32” 32”/2= 16” 2π(16”)= 100.5309649”

64” 64”/2= 32” 2π(32”)= 201.0619298”

Pi also connects the diameter or radius of a circle with the area of a circle by the formula. Archimedes discovered that the area of the circle is between the circumscribed and inscribed hexagons.

￼ (

Concentric Circles:

Concentric circles are when two or more circles centers coincide. Below is an example of concentric circles.

￼

Concentric circles have different radii but still have the same center so the circumference can still be found by￼. This proves that pi is the same for all size circles.

Pi is irrational:

Irrational numbers are numbers that cannot be expressed as...

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