How to find the surface area of a sphere?

[live4physics]

4 Pi r²

I know this formula, but anyone knows, how can I figure out (by methods, algebra) the sphere surface formula ?

Thanks

 

[arildno]

You might, for example, differentiate the formula for the volume of a ball with respect to the radius.

There are many other ways as well.

 

[live4physics]

You might, for example, differentiate the formula for the volume of a ball with respect to the radius.

Sorry, I did not say, but I would like not to use calculus for this, just geometry or algebra for demonstration.

For exemple:
the triangle area is:
$$A_{t} = \frac{b h} { 2 }$$
if I put a triangle inside a circle, where the top of it is the centre of circle, and I use infinity number of triangles, and with base near to zero, the sum of all bases is igual $$2 r \pi$$ and the height will be iqual the radius of circle, then, the area of circle will be the sum of area for all triangles, i.e.,

$$A_{c} = \frac{2 r \pi r } { 2 } = \pi r^2$$

Thank you very much.

 

[arildno]

Well, you might have a go at Archimedes' way of proving it, here's a link to a page describing that method:
http://www.maths.uwa.edu.au/~schultz/3M3/L7Archimedes1.html

 

[mathman]

Archimedes (although he didn't know it) invented integral calculus to get things like the calculation of pi by taking the limit of areas of regular polygons to get the area of a circle.

 

[HallsofIvy]

In general there is no way to determine the area under a curve or of a curved surface without using some kind of limit process, which is at least walking on the boundary of calculus.