In my math textbook there's a demonstration for the calculus of the circumference of a circle that involves regular polygons inscribed in the circle, but I don't get it. The book gives the following formula: $$l_{2n}=\sqrt{2R^2 – R\sqrt{4R^2-l_{n}^2}}$$ Where $l_{n}$ is the side of a regular n-sided polygon inscribed in a circle with R radius. Then, somehow using that formula, the books tells me that: $$l_{4}=R\sqrt2$$ But how do I reach that number? What was substituted for what?
And what's the use of that formula? Suppose, for example, that a square is inscribed in a circle. I know the radius and want to find the side of the square. How would I apply the formula? It doesn't exist a polygon with 2 sides!
The book then applies the same formula to a polygon with 8 sides, then one with 16, then 32, then 64 and shows the larger the number of sides of a polygon, the closer its perimeter is to the circumference. I can understand that, but I don't understand how it can find a number such as $R\sqrt2$ for the side of square without knowing the side of a hypotetical 2-sided polygon.
I hope I've made my question clear.
Best Answer
Probably the author does not intend to say that the expression for $l_4$ follows from the formula. Actually, it does, sort of. We have $l_2=2R$, and substitution in the formula gives $l_4=R\sqrt{2}$. We have used the formula starting from a "degenerate" polygon. It happens to work, but is liable, for good reason, to cause confusion.
However, $l_4$ can be computed directly, using basic geometry. The diagonal of the square inscribed in the circle is $2R$. So if $s$ is the length of a side of the square, then by the Pythagorean Theorem $$s^2+s^2=(2R)^2=4R^2,$$ so $s^2=2R^2$ and therefore $s=R\sqrt{2}$.