Pi, Circles and Spheres!

If you have done Maths to a secondary school level at some point you will have encountered circles, spheres and pi. But this information may have passed over your head or you never really went into the details of why these things work.

All circles have the amazing property that their circumference and diameter are related by a ratio, this ratio is known as pi. The interesting property about this number is that it is transcendental, this means that the decimal expansion is non-repeating at any stage and continues on for infinity (or in more formal mathematical terms, there is no polynomial with rational coefficients where π is a root). This property in itself is beautiful.

Pi has in fact been calculated to insane precision, to roughly ten trillion decimal places! This was obviously done using a computer, no human would try to take on this task (I sincerely hope). It is not very necessary to have such an accurate representation of π as if it is calculated to 11 decimal places you can accurately estimate the circumference of any circle that fits inside of the Earth to one millimetre. And if π is represented to 39 decimal places you can estimate the circumference of any circle inside the observable universe to the radius of one hydrogen atom. So 10 trillion decimal places is slightly unnecessary, but that is one way to get your name in the history books.

If you have ever really put any thought to maths you will often come to the thought of "why?". Why do the formulas we are told and have to recite work? And unfortunately, schools very rarely pay attention to the why and just have you use formulae blindly. And this thought is often the case with the area of a circle. We all know that it is πr² but why is it that way? Like all maths that we use, it has been proved rigorously by many individuals.


Area of Circle:
Here is a rather simple proof of why the area of a circle is so. If we split a circle into an even number of pieces we can rearrange them into a shape that resembles a parallelogram.

All I have done with my circle of radius r is split it into 8 equal sectors and move them around so they interlock in a  parallelogram  like shape. The top of the  parallelogram  like object uses half of the chunks we have used, which is half of the circumference, which is πr. Again from GCSE Maths you may remember that the area of a  parallelogram is defined as base × height. As the sectors of the circle get smaller in width the parallelogram begins to take shape and the top and bottom of the shape begin to flatten out. 

So sectors of width infinitely small will create a perfect parallelogram of height r and base πr, therefore the area of this parallelogram (and thus the circle) is πr²! And that is why we have our concise formula for the area of any circle.


There is also a proof for why a the volume of a sphere is also correct, but this requires some knowledge of calculus. So if you do not know calculus perhaps do not continue reading, however you are more than welcome to and then try to begin to understand calculus.


Volume of Sphere:
A sphere can be thought of as revolving a semi-circle around the x-axis. To do this consider a semi-circle circle with a centre at the origin and a radius of r, this will have an equation:

Any cross-section of the sphere derived by revolve the semi-circle, y, is a circle with a radius of y and an area of πy²! If we slice the sphere up into a infinite number of slices of infinitesimal thickness then the area of each slice is also it's volume (as it has an infinitely small thickness, essentially 0). So this means that the volume of each slice is πy², if we add the volumes of every single slice then we get the total volume, which is that of a whole sphere!


This means that we are talking about integrating πy² between r and -r to find the total area, and as area is the same as the volume for each slice the total volume.

And that's all! We have proved that formulae for the area of the circle and volume of a sphere are in fact correct. If there is anything at all that I have done here please do not hesitate to leave a comment and I will respond and explain to the best of my abilities.

Infinite Circles Problem

I recently encountered a mathematical problem from NRich Maths about inscribed circles in an equilateral triangle and it really intrigued and after a little bit of intense thought I managed to figure it out.


For those of who wish to know what the problem is without clicking the link:

A circle of radius 1 cm is inscribed in an equilateral triangle. A smaller circle is inscribed at each vertex touching the first circle and tangent to the two 'containing' sides of the triangle. This process is continued ad infinitum...
 

What is the sum of the circumferences of all the circles?

What is the sum of their areas?

Adding all the circumferences or adding all the areas, which sum grows faster?

Now this might not be immediately obvious as the best way to approach this problem, so we need to think about what we know, what we need to know and the best way to approach this.


What We Know:

1. The radius of the largest circle is 1cm.
2. All of the triangles angles are 60° as the triangle is an equilateral.
3. The area of the first circle is π, the circumference is 2π.

What We Need To Know:

1. The ratio of the radii from each circle to the next.
2. The height of the triangle.
3. The area of all the circles.
4. The circumference of all the circles.

The height of the triangle may seem like a bit of a strange necessity, but if you know the diameter of the main circle (2cm) then it helps to know what the sum of the diameters of all of the circles will be (height-2).

Now if the the radius were arranged so it was at a right angle to the triangle and a line was drawn from the centre of the largest circle the corner of the side the radius touches the angle would be half of the original angle which is 60° so the new angle is 30°. This is hard to picture but an angle will help that.

Now we have two angles and one side, so we can use the Sin rule to find the size of the line from the centre of the circle to the corner of the triangle.

This means that the radius of the largest circle plus the diameters of all of the other circles is 2. So the height of the triangle 2 plus the radius of the larger circle, which equals 3. It also means that the sum of the diameters of all of the other circles equals 1 too.

Now we can begin to actually tackle the problem of the sum of the circumferences of all the circles. We already know that the circumference of the first circle is 2π, if the sum of all the diameters of the other circles in one line is 1 we can see that the area is then π, but we still have two other sets of circles. So we have the total circumference of 2π+π+π+π which means the total circumference is 5π. Problem one solved.

The second problem is slightly more awkward as the radius is not as easy to find and although the way I am about to explain does work in may not be the most efficient, but it does work and it utilises some very nice Core 2 techniques.

As you may have noticed, there will be an infinite number of circles going into any of the corners (this is caused by the curved shape of a circle against the straight side of the triangle). If we exclude the large circle then the sum of the diameters of into one corner is 1.

The fact that the triangles always get smaller, means that the rate at which they 'increase' is less than 1, we will call this ratio 1/n, the radius of the second triangle will also be 1/n because of the fact that the first radius is 1.

We know that the sum of the diameters equals 1, which means that the sum of 2*radius is also equal to 1. We also know the first term of this series (1/n), the ratio of the series (1/n) and the sum of the series (1). As our ratio is less than 0 we can use the formula covered in C2 for that:

Using that we can rearrange to find what n equals and thus find the ratio. I have included the original equilateral triangle image along with some labelling to help to explain my notation.

So we have that the ratio from radius to radius is 1/3, so to find the sum of the area of all the triangles we must use the sum of an infinite series again. Given that the first term is π (from πr^2 and r = 1), the ratio is r^2, which gives 1/9. We have three of the series so we will times the sum of these by 3, but then we have included the largest triangle three times, so we must subtract this two times (-2π).

This means that the area of all the circles is 11π/8, problem two solved.

The last problem is considerably easy to handle, it simply asks which sum grows faster, this is the one that has a larger ratio. Well the ratio of the area is 1/9, whereas the circumference is 2/3 (r is 1/3, but we want twice this). So this means the circumference increases faster.

This problem really is a lovely one, it combines some relatively simple maths in an advanced form, pieces them all together and leaves you to solve the puzzle. Maths is fun. Maths is really, really fun!

I realise I may have explained fair chunks of this poorly, it is very difficult to convey what is happening and without being in front of you. So if you are left with any questions as to what I have done, or why, simply leave a comment and I will explain or email me at lewis.mead@eloquentmath.com for more information.

Also to let you know, I will be completing a Core 2 revision guide pretty soon (give me a week or so), so keep checking back here for updates on that.

Python: Mathematical Terminology

This will be my first actual "mini-python" lesson, I'm going to assume you've read my previous post on Python, if you haven't please read it: an introduction to Python. Got everything you need downloaded? Then I'll begin.

Again this is just going to cover the absolute basics, but they are incredibly vital to know. Maths is the core behind the majority of programs, regardless of what you want to program you're bound to come across maths at some point when programming (especially with my posts, seeing as I am mathematical nerd).

Most of the mathematical characters are fairly obvious; "*" meaning times, "+" meaning add, "-" meaning minus, "/" meaning divide and "=" meaning equals. Some of the less obvious ones are to the power of, "**" and to show the remainder when divided is "%". But once you can remember these it's very easy to apply these.

Another important thing to remember is that if you want to 'print' the answer to mathematical you do not use quotes, if you just want to show the expression you do.

print "2 + 2 =", 2+2

Will show:

2 + 2 = 4

Simple right? Just remember to separate the different strings you want to print with a comma. Now let's create a basic program using these concepts.

print "The area of a circle with radius 15 is", 3.142 * 15**2

That would show:

The area of a circle with radius 15 is 706.85834715

Try this out with different numbers, expressions and signs. Only through doing this yourself will you begin learning. In the next lesson I will bring in variables and begin coding a small working program where the user inputs data.