Sphere Mathimatics

Here is the complete proof, I added the previous proof of the 4/3 figure into it......



Proving the Volume formula for a Sphere, 4/3 * pi * r^3

For the following equations, the heights for the cone and cylinder, h, will be the same measurement of the radius r.

Cone Volume = 1/3 * pi * r^3

Cylinder Volume = pi * r^3

The above are accepted and understood, and will not be proved here.

(Refer to the attached image).

Okay, here is the formula for volume of a sphere = 4/3 * pi * r^3

Working backwards, the 4/3 is actually 2 * (2/3 * pi * r^3).

The (2/3 * pi * r^3) is the volume of a hemisphere, which is, as you know, half of the sphere.

The 2/3 part comes from the area of a hemisphere is volume of a cylinder - the volume of a cone (hemisphere has radius r, which is equal to height of cylinder and cone).

The idea to be proven is that the volume of the hemisphere (one half of the sphere) can be figured by calculating the volume of a proportional cone and proportion cylinder, with the height of each being equal to the radii.

The image is a vertical cross section of the 3d cones, sphere and cylinder. Looking at the picture, the square represents the section of the cylinder, the crossed X represents the two cones, and the circle represents the sphere.

Looking at the three solids, take a horizontal cross-section between C and E. This cross-section intersects the three solids, giving three concentric circles, with the radii of CD (cone), CB (sphere), and CE (cylinder).

The areas of these circles are proportional to the squares of the radii.
Area of a circle = pi * r^2

It is now observed that the squared radius of the cylinder has to be the squared radius of the cone plus the squared radius of the sphere.

Now for connecting the dots ...

(The following are lines; they are represented with the omitted -- on top).

CE = AF = AB

CA = CD, so CA^2 = CD^2

CA^2 + CB^2 = AB^2, and we see above that AB = CE, so AB^2 = CE^2

So therefore, since CA^2 = CD^2

CD^2 + CB^2 = CE^2

So now a proportion is clearly established.
Circle (cone) + circle (sphere) = circle (cylinder), so,
Circle (sphere) = circle (cylinder) - circle (cone).

The Volume of Cylinder (pi * r^3) - Volume of Cone (1/3 * pi * r^3) = 2/3 * pi * r^3, so,

Multiplying by 2 (for two hemispheres) gives 4/3 * pi * r^3.

:p

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So bugz,

whatchya tink?

i've briefly looked at it and it looks good. but i want to take a closer look tomorrow (with paper and pen and such). but good work!

I made a notation error:

Quote:

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Working backwards, the 4/3 is actually 2 * (2/3 * pi * r^3).

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It should read :

Code:

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Working backwards, the 4/3 is actually (2 * 2/3) * pi * r^3.

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Nice proof, Jared!:D

I tried to develop a similar proof for the surface area of a sphere to the one that I had for the volume of a sphere, and there's something wrong with it, because the result is off. Here it is:

Draw in an infinite amount of "vertical' circumferences equidistant at the equator of the sphere all passing through two points, one at the top and one at the bottom. Then draw in the "horizontal" circumference at the equator of the sphere. Now there is an infinite amount of approximations of isosceles triangles each bounded by two (quarters of) vertical circumferences and (a small section of) the cirumference at the equator. Let the base of each of the triangles be b, where b approaches 0. The height of each of these triangles approaches a quarter of any circumference, which is pi*r/2. The total number of triangles is 2*(2*pi*r/b), and the area of each triangle is (1/2)*b*(pi*r/2). By this logic the surface area of the sphere should be 2*(2*pi*r/b)*(1/2)*b*(pi*r/2)=pi^2*r^2.

Why am I off by a factor of 4/pi?

Quote:

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Originally posted by Jared
Here is the complete proof, I added the previous proof of the 4/3 figure into it......

Now for connecting the dots ...

(The following are lines; they are represented with the omitted -- on top).

CE = AF = AB

CA = CD, so CA^2 = CD^2

CA^2 + CB^2 = AB^2, and we see above that AB = CE, so AB^2 = CE^2

So therefore, since CA^2 = CD^2

CD^2 + CB^2 = CE^2

So now a proportion is clearly established.
Circle (cone) + circle (sphere) = circle (cylinder), so,
Circle (sphere) = circle (cylinder) - circle (cone).

The Volume of Cylinder (pi * r^3) - Volume of Cone (1/3 * pi * r^3) = 2/3 * pi * r^3, so,

Multiplying by 2 (for two hemispheres) gives 4/3 * pi * r^3.

:p

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You lost me again. :confused:

Using the attached image on my earlier post, evaluate each mathematical statement.

Each statement will build upon a previous statement to lead to the conclusion. I broke it down into simple statements, so each one shouldn't be too great of a leap of understanding.

Give me a specific line where you are stuck, and I will explain it.


:)