In the Frame beside this one, you will see a dynamic document that displays an animation which shows an ellipse of the general formula:

Here if the "a" term is larger than the "b" term, we have an ellipse with its major axis oriented horizontally.

When the "a" term just equals the "b" term, then both axis are the same size, hence there is no major axis,- and the ellipse becomes a perfect circle.

Finaly, when the "a" term is less than the "b" term, we get an ellipse with the major axis oriented vertically.

As used in the above equation, the "a" term could be used to determine the ellipse's eccentricity, e by calculating the ratio between it and the "b" term. Here when a > b, e > 1
When a = b, e = 1 , and when a < b, e < 1 .

The "a" term here should not be confused with the "a" term often used to determine the eccentricity of an ellipse where "a" is the distance between the two focal points along the major axis.

Our animation begins in quadrant I where both the "h" term and the "v" term are positive. When the ellipse has been transformed into a circle by reducing the "a" term to equal the "b" term, we can see that the circle's lower limit (point P1) is positive. The circle's upper limit (point P4) is more positive.

Likewise both the circle's maximum (point P2) and minimum (point P3) are positive. Our circle is moved over to the graph's Y axis by reducing the "h" term to zero. Then the circle is moved down to the X axis by reducing the "v" term to zero.

Now our circle is centered on the origin. It is easy to note the circle's minimum (point P3) has become negative as has the circle's lower limit (point P1). We can redefine the circle's upper limit as +R, and its lower limit as -R since P1 is located at (-R, 0) and P4 is located at (+R, 0).

We have now reduced our general equation for an ellipse to the specific equation of a circle centered upon the origin:

By simple algebra, we can rearrange this equation to:

The animation now shifts our point of view so that instead of looking at the face of our circle, we are looking at the top edge of the circle. We can place a coin on a table and look straight down on it from above to see the same effect.

If we were to spin the above mentioned coin on its edge, that coin would enclose a spherical volume. By rotating our flat circle about its Y axis we generate Z axis values that did not exist before, turning our circle into a sphere.

In the adjacent frame you can animate a full circle by selecting +R for the upper limit, and -R for the lower limit.

Or you can animate a half circle by selecting +R for the upper limit again, only setting the lower limit at 0 instead of at -R.

In either case we end up with a full sphere because we rotate the full circle through radians, while we rotate the half circle through 2 radians.

We can calculate the
volume of revolution through the mathematical process of integration using another general formula:

For a simpler calculation, it would be advisable to move the lower limit from -R to zero:

By substituting the value of Y² we arrived at while discussing the circle into the preceding equation we have:

Now we have the equation we need. Integrating the above equation yeilds:

Applying the limits to the expression inside the brackets in the previous equation gives us:

Simplifying the terms within the brackets brings us to:

Expressing the R³ term in thirds we get:

Combining like terms:

Removing the brackets finally gives us our well known formula for the volume of a sphere:

While this may seem like the hard way to solve the volume of a sphere, the method works for litterally any shape at all. It also shows how mathematicians can move a two dimensional object into the 3rd dimension. The same method allows us to move three dimensional objects into the 4rth dimension. This is the means by which we can look into N dimensional "spaces."

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