Basel's Problem
$$1+\frac{1}{4}+\frac{1}{9}+\cdots = \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi ^2}{6}$$
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Area of the Circle - Alternative
Here's an elementary way to find the area of a circle although I'm sure many of you are already familiar with it. Slice up the circle into 4 parts and stack them up as follows.
Viviani's Theorem
Viviani's Theorem states that for any point $\mathcal{P}$ inside an equilateral triangle $\mathcal{ABC}$ the sum of the lengths of perpendiculars drawn from P on the sides is equal to the length of the altitude. Draw equilateral triangles from that point onto the sides of the triangle such that these 3 perpendiculars are the altitudes of the equialteral triangles.Now rotate the triangles as shown.Now it can be easily seen that the sum of the 3 perpendiculars=perpendicular of the $\Delta ABC$
The Triangle Inequality
Without loss of generality we may assume $a\leq b\leq c$. It is quite obvious that $$a+c\geq b\qquad a+b>c$$ Rotate $\mathcal{BC}$ about $\mathcal{B}$ and $\mathcal{AC}$ about $\mathcal{A}$. Now as we can see in the above diagram $$a+b>c$$ NOTE: $a+b=c$ would imply that the triangle is degenerate.