Math Is Fun Forum

Happy3690

Member

Registered: 2015-11-07

Posts: 2

Surface area of a sphere???

The surface area formula is 4pir^2

Yet you could think of it as the sum of all the circumference

Or the integral of 2pir*2. The integral of 2pir is. 2pir^2/2 this multiplied by two gives 2pir not 4 pir. Help!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Surface area of a sphere???

Hi;

It is a little more complicated than that.

https://www.youtube.com/watch?v=J42wNwCh0Mk

http://www.mathalino.com/reviewer/deriv … ntegration

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Surface area of a sphere???

hi Happy3690

Welcome to the forum.

If you tried to add up all possible circumferences like this there is a problem with the process.  All the circumferences overlap at the top and bottom of the sphere so you are counting the same bits more than once.  Also if you think about lines of longitude on the Earth for example; they are close together at the poles but much wider apart at the equator.  So the summation does not count the surface area fairly.

You can think of the sphere as shrinking inwards an infinitesimal amount (by radius dr) .  So differentiating the formula for volume will generate an infinitesimally thin volume (the change of volume with respect to radius) which would be the surface area:

As the volume formula can also be found (using integration of thin discs), this method is considered acceptable.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Maximoff

Member

Registered: 2015-10-23

Posts: 10

Re: Surface area of a sphere???

Hi,

Basically, you would like to understand why the surface area for sphere is 4*pi*r^2 right.
As far we know, sphere perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.
So basically, it like super-circle but in way of three dimensional.

As we recall, the circumference of circle is given by 2*pi*r where it define the distance all around the circle.
Given that sphere is super-circle that shaped like round ball, we can assume that the formula of circumference multiplied with 2*r (due to the fact that the center of ball is connected with many line of radius intersect with all the point along the distance around the circle).

As conclusion, why the formula for surface area of sphere is given by 4*pi*r^2 is due to the fact of circumference of circle times with diameter of the circle or 2*r .

*** This is only my point of view. Do check with the website provided by bobbym. Thanks

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Surface area of a sphere???

hi Maximoff

For a plane rectangle you would multiply the length by the width to get the area.  So I can see what you are trying to do here.  But a circumference is the limit of a sector, where the angle (at the pole) diminishes to zero.  You can see in this diagram that sectors do not have a constant width, so I do not think this approach is valid. 

You'll have to justify that your method is the sum of rectangular portions, and I don't think it is.  Sorry. 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob