matematicas visuales visual math

In the initial position of the applet it represents a circle, and when you move the cursor the vertical position of a segment changes.

When the vertical position (x) changes we want to calculate the segment a:

Sections in a sphere and Geometric mean: right triangle and chord | matematicasVisuales

Because our goal is to calculate the surface of the section of the sphere when we change the distance from the center of the sphere to the section. You can see and rotate the sphere clicking and dragging over the applet.

Sections in a sphere and Geometric mean: surface area of a section | matematicasVisuales

Using the Pythagorean Theorem we can calculate the radius of the section:

Sections in a sphere and Geometric mean: radius of the section | matematicasVisuales

Then the surface of the section of the sphere is:

Sections in a sphere and Geometric mean: calculation of the surface area of a section| matematicasVisuales

Then our goal is achieved, but we can have another approach to calculate the chord (or radius, a) without using the Pythagorean Theorem. And this approach will permit us to use similarity of triangles and to talk about the Geometric Mean.

Sections in a sphere and Geometric mean: similarity of triangles and the Right Triangle Altitude Theorem | matematicasVisuales

We have three right triangles that are similar. We are interested in two of them:

We can write the proportion (we want to know the value of a):

Then, the value of a is:

We say that a is the Geometric Mean of two numbers, b and c. This result is also called the Right Triangle Altitude Theorem.

The geometric mean of two positive numbers is related with the arithmetic mean:

Sections in a sphere and Geometric mean: The geometric mean is equal to the arithmetic mean when the numbers are equal | matematicasVisuales

When are the geometric mean equals to the arithmetic mean?

Coming back to our initial subject:

Sections in a sphere and Geometric mean: calculating the radius of a section using geometric mean | matematicasVisuales

We are going to use this result in two interesting applications of Cavalieri's Theorem: How to calculate the volume of a sphere and Two Surprising Theorems on Cavalieri Congruence one article by Howard Eves in which he constructed a tetrahedron and he used Cavalieri's Theorem to calculate the volume of a sphere.

You can play with these two applets about cross-sections of a sphere:

Sections of a sphere: volume | matematicasVisuales

Sections of a sphere: two parts | matematicasVisuales

REFERENCES

Howard Eves, mathematician and historian of Mathematics, received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence.

MORE LINKS

The volume of the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
Sections in Howard Eves's tetrahedron
In his article 'Two Surprising Theorems on Cavalieri Congruence' Howard Eves describes an interesting tetrahedron. In this page we calculate its cross-section areas and its volume.
Cavalieri: The volume of a sphere
Using Cavalieri's Principle we can calculate the volume of a sphere.
Kepler: The Area of a Circle
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
Pythagoras Theorem: Euclid's demonstration
Demonstration of Pythagoras Theorem inspired in Euclid.
Plane developments of geometric bodies: Tetrahedron
The first drawing of a plane net of a regular tetrahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
Archimedes and the area of an ellipse: an intuitive approach
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. We can see an intuitive approach to Archimedes' ideas.
Archimedes and the area of an ellipse: Demonstration
In his book 'On Conoids and Spheroids', Archimedes calculated the area of an ellipse. It si a good example of a rigorous proof using a double reductio ad absurdum.
Archimedes' Method to calculate the area of a parabolic segment
Archimedes show us in 'The Method' how to use the lever law to discover the area of a parabolic segment.
Campanus' sphere and other polyhedra inscribed in a sphere
We study a kind of polyhedra inscribed in a sphere, in particular the Campanus' sphere that was very popular during the Renaissance.
Leonardo da Vinci:Drawing of a SEPTUAGINTA made to Luca Pacioli's De divina proportione.
Leonardo da Vinci made several drawings of polyhedra for Luca Pacioli's book 'De divina proportione'. Here we can see an adaptation of the Campanus' sphere.