1. ## Why is area of circle pi x radius^2?

I know that the area of a circle is pi x radius^2 but I would like to know why is area of circle pi x radius^2. In other words, how did mathematicians conclude that the area of a circle is pi X radius^2?

Thanks,

Ron

2. Originally Posted by rn5a
I know that the area of a circle is pi x radius^2 but I would like to know why is area of circle pi x radius^2. In other words, how did mathematicians conclude that the area of a circle is pi X radius^2?

Thanks,

Ron
By similarity arguments it can be shown that the ratio of the areas of circles of radii $\displaystyle r_1$ and $\displaystyle r_2$ is $\displaystyle r_1^2/r_2^2$

By definition $\displaystyle \pi$ represents the area of a circle of radius $\displaystyle 1$, and the rest follows.

CB

3. Here is one way to do it. Imagine dividing the circle into n "wedges". Now, line the wedges up so that the first has its "tip" upward, the second downward, etc., reversing the direction on each one. We get approximately a rectangle with the arcs of the circle forming both upper and lower edges, with the approximation becoming better for larger n. That "rectangle" has the radius of the circle as height and half the circumference as width (since the circumference formed both top and bottom). Since the circumference is $\displaystyle \pi d= 2\pi r$, half the circumference is $\displaystyle \pi r$. The area of the "rectangle", and so the area of the circle, is "height times width" $\displaystyle = r(\pi r)= \pi r^2$. In the limit, as n goes to infinity, that "approximation" becomes exact. Since the true area of the circle does not depend on n, that formula gives the true area.

Notice that Captain Black is defining "$\displaystyle \pi$" to be the area of a circle of radius 1 while I am taking the definition of "$\displaystyle \pi$" as the ratio of the circumference of a circle to its diameter.

4. Originally Posted by HallsofIvy
Here is one way to do it. Imagine dividing the circle into n "wedges". Now, line the wedges up so that the first has its "tip" upward, the second downward, etc., reversing the direction on each one. We get approximately a rectangle with the arcs of the circle forming both upper and lower edges, with the approximation becoming better for larger n. That "rectangle" has the radius of the circle as height and half the circumference as width (since the circumference formed both top and bottom). Since the circumference is $\displaystyle \pi d= 2\pi r$, half the circumference is $\displaystyle \pi r$. The area of the "rectangle", and so the area of the circle, is "height times width" $\displaystyle = r(\pi r)= \pi r^2$. In the limit, as n goes to infinity, that "approximation" becomes exact. Since the true area of the circle does not depend on n, that formula gives the true area.

Notice that Captain Black is defining "$\displaystyle \pi$" to be the area of a circle of radius 1 while I am taking the definition of "$\displaystyle \pi$" as the ratio of the circumference of a circle to its diameter.
..or defining it to be the circumference of a circle of unit diameter (but you still need a similarity argument to prove that the circumference is proportional to the radius).

CB

5. once Pi is defined in radians you can integrate a quadrant of a circle in polar coords to get Pi r^2

6. Originally Posted by Krahl
once Pi is defined in radians you can integrate a quadrant of a circle in polar coords to get Pi r^2
I don't think that's quite what the OP wanted!