Dojo
The Spinning Sphere - Trivial Math Proofs - AoPS activity - Contact

My name is Dojo and I currently am 14, and live in Washington.

My interests are math, technology, solving rubiks cubes, cello, piano, composing, track, cross country and tennis, just to name a few.

The Spinning Sphere

Yes yes,[big voice] I am the creator of the almighty spinning sphere!!! [/end big voice] Yeah well anyway, for anyone interested, I have created a gallery of these spheres: My Gallery (My gallery is now the Animation Studio)

Trivial Math Proofs

Suggest your trival proofs you want here.

Equilateral Triangle Area

Proof that the area of an equilateral triangle with side length $s$ is $\dfrac{s^2\sqrt {3}}{4}$ :

Let's say that there is an equilateral triangle that has a side length of $s$ . We can then draw the following figure:

$[asy] draw((0,0)--(1,sqrt(3)),linewidth(1)); add(pathticks((0,0)--(1,sqrt(3)),1,green+linewidth(1))); draw((2,0)--(1,sqrt(3)),linewidth(1)); add(pathticks((2,0)--(1,sqrt(3)),1,green+linewidth(1))); draw((2,0)--(0,0),linewidth(1)); add(pathticks((2,0)--(0,0),1,green+linewidth(1))); label("$s$",(1,0),S); [/asy]$

Now let's figure out the altitude so we can complete the triangle area forumla of $\dfrac{bh}{2}$ :

$[asy] draw((0,0)--(1,sqrt(3)),linewidth(1)); add(pathticks((0,0)--(1,sqrt(3)),1,green+linewidth(1))); draw((2,0)--(1,sqrt(3)),linewidth(1)); add(pathticks((2,0)--(1,sqrt(3)),1,green+linewidth(1))); draw((2,0)--(0,0),linewidth(1)); add(pathticks((2,0)--(0,0),1,green+linewidth(1))); label("$s$",(1,0),S); draw((1,0)--(1,sqrt(3)),dashed+linewidth(1)); [/asy]$

We can now use the pythagorean theorem to find the length of the altitude:

$[asy] draw((0,0)--(0,sqrt(3))--(1,0)--cycle,linewidth(1)); draw(rightanglemark((0,sqrt(3)),(0,0),(1,0)),red+linewidth(1)); [/asy]$

Since we know that this is a $30 - 60 - 90$ triangle, we can use proportions to find the altitude $a$ in terms of side lenth $s$ :

$\begin{align*} \dfrac{2}{\sqrt {3}} & = \dfrac{s}{a} \\ \sqrt {3}s & = 2a \\ \dfrac{\sqrt {3}}{2}s & = a \end{align*}$ (Error compiling LaTeX. Unknown error_msg)

Now plugging this altitude into the triangle area forumla gives us:

$\dfrac{\frac {\sqrt {3}}{2}s\times s}{2} = \dfrac{\frac {s^2\sqrt {3}}{2}}{2} = \boxed{\dfrac{s^2\sqrt {3}}{4}}$

Proof can be found on this post of my blog.

Diagonal Forumla

Proof that the number of diagonals in a polygon with $n$ sides is $\dfrac{n(n-3)}{2}$ :

First, lets see the hexagon:

$[asy] size(200); for(int i=0; i<6; ++i) for(int j=i+1; j<6; ++j) draw(dir(60*i)--dir(60*j)); [/asy]$

If you count carefully, you'll see that there are 9 diagonals.

Now we need to see how we can derive a forumla for the number of diagonals.

For any polygon with $n$ sides, we see that there are $n$ vertecies. To create a diagonal, we need one other point, which can be selected from a pool of $n - 3$ points. We must exclude 3 points because the point connecting to the point itself doesn't count as a diagonal, and connecting to the 2 adjecent points don't count because they have already been "drawn in" as the sides of the polygon. We would then assume that there are $n(n - 3)$ diagonals, right? Wrong. Let's say that two of the points are $A$ and $B$ . Using the above method, both the diagonals $AB$ and $BA$ would be counted. Therefore, we must divide the forumla by 2, giving us the diagonal forumla:

$\boxed{\dfrac{n(n - 3)}{2}}$

Proof can be found on this post of my blog.

Handshake formula

Proof that the number of handshakes occuring in a group of $n$ where each person shakes each other's hand is $\dfrac{n(n-1)}{2}$ .

This proof is similar to the diagonal proof, with one exception. Each "vertex" is replaced by a person and people can shake hands to adjacent people. Since each person will shake hands with $n-1$ people (Everyone except themselves) and there are $n$ people, we get a doubled $n(n-1)$ handshakes. Once again, we must discount the doubled handshakes occuring between say person A and B. Therefore, we have $\boxed{\dfrac{n(n-1)}{2}}$ .

Proof can be found on this post of my blog.

AoPS activity

AoPS activity.

Contact

Some ways you can reach me:

Email. dojothegreat@gmail.com
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Contents

The Spinning Sphere

Trivial Math Proofs

Equilateral Triangle Area

Diagonal Forumla

Handshake formula

AoPS activity

Forums I moderate

Blog

Wiki pages I created

Contact