# What is the official proof (if there is any) for the area of a circle of radius 'r'?

What is the official proof (if there is any) for the area of a circle of radius 'r' ?

I remember in my school days they simply told that area of a circle of radius 'r' is $\pi*r^{2}$.

The teacher also told, $\pi$ = $\frac{circumference}{diameter}$

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Do you know calculus? – Calvin Lin Apr 12 '13 at 2:36
– Amzoti Apr 12 '13 at 2:38
do you have some other definition for $\pi$? Or is this asking why it should be proportional to $r^2$ in the first place? – Robert Mastragostino Apr 12 '13 at 2:41
They never prove that the area of a circle is $\pi*r^{2}$ in any text book (for e.g., the text book of CLASS X). They say it is $\pi*r^{2}$. – Rajesh K Singh Apr 12 '13 at 3:25
@ Calvin Lin : circle is introduced in class 8 if i'm right. calculus takes the $\pi$ from there. I'm talking of basics. If basics are not convincing how do we proceed further. – Rajesh K Singh Apr 12 '13 at 3:30

This also gives the following pleasant heuristic. Regular polygons can be made of identical triangles from their center. For a 'nearly infinite'-sided polygon inscribed in a circle of radius $r$, each of the triangles will have altitude essentially $r$. The circumference of the circle is $2 \pi r$, so putting all these triangles together will give a triangle with are $\frac{1}{2} \cdot (r) \cdot (2 \pi r)$ from $\frac{1}{2} \cdot \text{height} \cdot \text{base}$, heuristically.