# The volume/SA of a sphere

• February 9th 2010, 04:14 PM
Masterthief1324
The volume/SA of a sphere
I decided to abide by logic in order to reconstruct the formula for the volume of a sphere. Juxtapose against the actual formula, my formula does not match.

Why is it so?

My reasoning:
Let r be the radius of a circle.

1. I first use 'r' to find its 90 degree sector (is this how you would express finding a particular sector of a circle?), then I have area 'x'.

$\frac {90}{360} = \frac {x}{(\pi)r^2}$ ;where x is the area of the sector.

2. If I take x, or its equivalent fraction, and multiply it by the circumference of the circle with the radius of 'r' I should get the volume of half the sphere.

$\frac {90}{360} = \frac {x} {(\pi)r^2}$ simplifies to
$\frac {(\pi)r^2}{4}$

Muliply that by:

$
\pi2r
$

and you have:

$
\frac {\pi^2r^3}{2}
$

3. Multiply the fraction by 2 and it does not match up with the volume for sphere. Why not?
• February 9th 2010, 05:08 PM
skeeter
Quote:

Originally Posted by Masterthief1324
I decided to abide by logic in order to reconstruct the formula for the volume of a sphere. Juxtapose against the actual formula, my formula does not match.

Why is it so?

My reasoning:
Let r be the radius of a circle.

1. I first use 'r' to find its 90 degree sector (is this how you would express finding a particular sector of a circle?), then I have area 'x'.

$\frac {90}{360} = \frac {x}{(\pi)r^2}$ ;where x is the area of the sector.

2. If I take x, or its equivalent fraction, and multiply it by the circumference of the circle with the radius of 'r' I should get the volume of half the sphere.

Why not?

because this idea is not true.

I refer you to the second theorem of Pappus ...

Pappus's Centroid Theorem -- from Wolfram MathWorld
• February 10th 2010, 02:09 AM
HallsofIvy
You say "abide by logic" and then you argue by analogy, not by logic.

Logic, if used correctly, always works. Argument by analogy often does not.
• February 10th 2010, 04:44 AM
Masterthief1324
Quote:

Originally Posted by HallsofIvy
You say "abide by logic" and then you argue by analogy, not by logic.

Logic, if used correctly, always works. Argument by analogy often does not.

Hence analogy is the weakest form of argument. Are you saying there is one real logic? If so then what am I reasoning by? Analogy?
• February 10th 2010, 04:53 AM
Masterthief1324
Quote:

Originally Posted by skeeter
because this idea is not true.

I refer you to the second theorem of Pappus ...

Pappus's Centroid Theorem -- from Wolfram MathWorld

Very interesting. Thank you for the referral.