matematicas visuales visual math
Pythagoras Theorem: Baravalle demonstration

We can play with two previous interactive applications to 'see' demonstrations of the Pythagorean Theorem. One is inspired in Euclid (although Euclid did not have a dynamic approach) and the other uses a tiling.

Pythagoras Theorem: Euclid's demonstration
Demonstration of Pythagoras Theorem inspired in Euclid.
Pythagoras' Theorem in a tiling
We can see Pythagoras' Theorem in a tiling. It is a graphic demonstration of Pythagoras' Theorem we can see in some floor made using squares of two different sizes.


The Pythagorean Theorem (or Theorem of Pythagoras) is one of the most famous theorems of Mathematics. It is a property of right-angled triangles.

Then the Theorem of Pythagoras states that the square on the hypotenuse equals the sum of the (areas of the) squares on the other two sides.

Theorem of Pythagoras, Pythagorean Theorem: Baravalle demonstration | matematicasvisuales


In this page we can interact with another dynamic and interactive demonstration of the Pythagorean Theorem devised by Hermann Baravalle (1945).

The main idea is that if a parallelogram is changed by a shearing (preserving its base and altitude), its area remains constant.

Theorem of Pythagoras, Pythagorean Theorem: Baravalle demonstration | matematicasvisuales
Theorem of Pythagoras, Pythagorean Theorem: Baravalle demonstration | matematicasvisuales
Theorem of Pythagoras, Pythagorean Theorem: Baravalle demonstration | matematicasvisuales

And if we consider a movement (a translation in this case) of a parallelogram the area does not change.

Theorem of Pythagoras, Pythagorean Theorem: Baravalle demonstration | matematicasvisuales

Another shearing:

Theorem of Pythagoras, Pythagorean Theorem: Baravalle demonstration | matematicasvisuales
Theorem of Pythagoras, Pythagorean Theorem: Baravalle demonstration | matematicasvisuales

This is a demonstration of the Pythagorean Theorem:



REFERENCES

Euclides, The Elements
Alexander Bogomolny, Cut the Knot. Pythagorean theorem.
H.S.M. Coxeter, 'Introduction to Geometry', John Wiley and Sons, Second edition, pp. 8-9.
John Stillwell, "Mathematics and its History", Springer-Verlag, New York, 2002.
Martin Gardner, 'Sixth Book of Mathematical Diversions from "Scientific American"'. Scribner, 1975.
Eli Maor, "The Pythagorean theorem: a 4000-year history", Princeton University Press, United States of America, 2007.
F.J. Swetz and T.I. Kao, "Was Pythagoras chinese?", The Pennsylvania State University Press, United States of America, 1977.

MORE LINKS

Central and inscribed angles in a circle
Central angle in a circle is twice the angle inscribed in the circle.
Sections in the sphere
We want to calculate the surface area of sections of a sphere using the Pythagorean Theorem. We also study the relation with the Geometric Mean and the Right Triangle Altitude Theorem.
Standard Paper Size DIN A
There is a standarization of the size of the paper that is called DIN A. Successive paper sizes in the series A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension.
Drawing a regular pentagon with ruler and compass
You can draw a regular pentagon given one of its sides constructing the golden ratio with ruler and compass.
Cavalieri: The volume of a sphere
Using Cavalieri's Principle we can calculate the volume of a sphere.
The icosahedron and its volume
The twelve vertices of an icosahedron lie in three golden rectangles. Then we can calculate the volume of an icosahedron
The volume of the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
Volume of an octahedron
The volume of an octahedron is four times the volume of a tetrahedron. It is easy to calculate and then we can get the volume of a tetrahedron.
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.
Surprising Cavalieri congruence between a sphere and a tetrahedron
Howard Eves's tetrahedron is Cavalieri congruent with a given sphere. You can see that corresponding sections have the same area. Then the volumen of the sphere is the same as the volume of the tetrahedron. And we know how to calculate this volumen.
Kepler: The best proportions for a wine barrel
Studying the volume of a barrel, Kepler solved a problem about maxima in 1615.