matematicas visuales visual math

The tetrahedron is a regular pyramid. We can calculate its volume using a well known formula: The volume of a pyramid is one third of the base area times the perpendicular height.

But we are going to make a construction that will help us to deduce easily the volume of a tetrahedron. Kepler showed us how to do that. The idea is to build a tetrahedron inside of a cube.

Taking two diagonals of two opposite sides of a cube and attaching them properly we get a tetrahedron.

Volume of a tetrahedron: Kepler and the tetrahedron | matematicasVisuales
Volume of a tetrahedron: tetrahedron inside a cube | matematicasVisuales
Volume of a tetrahedron: The volume of a tetrahedron is one third the volume of the cube that constains it | matematicasVisuales
Volume of a tetrahedron: The volume of a tetrahedron is one third the volume of the cube that constains it | matematicasVisuales
Volume of a tetrahedron: The volume of a tetrahedron is one third the volume of the cube that constains it | matematicasVisuales

We want to see that the volume of the tetrahedron is one third of the cube that contains it.

Volume of a tetrahedron: the volume of the tetrahedron is one third of the cube that contains it | matematicasVisuales

If the tetrahedron edge length is 1 then the cube edge length w is:

We can write the volume of a cube with diagonal 1 (as a function of its wedge):

Volume of a tetrahedron: volume of a cube with diagonal 1 | matematicasVisuales

We can calculate this volume:

Volume of a tetrahedron: The volume of a tetrahedron is one third the volume of the cube that constains it | matematicasVisuales
Volume of a tetrahedron: The volume of a tetrahedron is one third the volume of the cube that constains it | matematicasVisuales

This cube is made of one yellow tetrahedron and four green pyramids.

The base of two of these pyramids is the base of the cube. Then the volume of two pyramids (using the formula of the volume of a pyramid: one third of the base area times the height) is one third of the volume of the cube.

Volume of a tetrahedron: The volume of a tetrahedron is one third the volume of the cube that constains it | matematicasVisuales

The volume of these four green pyramids is two thirds of the volume of the cube. Then the volume of the yellow tetrahedron is one third.

We can see that the volume of the tetrahedron is one third of the cube that contains it.

Then, the volume of a tetrahedron with edge length 1 is:

The volume of a tetrahedron with edge length a is:



This construction can be generalized to any parallelepiped and we get not regular "tetrahedra" .



Volume of a tetrahedron: tetrahedra inside a prism | matematicasVisuales

The volume of one of these tetrahedra is one third of the parallelepiped that contains it.



We can build a tetrahedron using modular origami and a cardboard cubic box.

Building polyhedra: cubic box and origami modular tetrahedron | matematicasVisuales
Building polyhedra: cubic box and origami modular tetrahedron | matematicasVisuales
Building polyhedra: cubic box and origami modular tetrahedron | matematicasVisuales

This is a template for a cubic box to download. The rectangle is the paper size you need to build a tetrahedron that fit inside the cubic box:

Building polyhedra: cubic box and origami modular tetrahedron | matematicasVisuales

You can see more examples of modular origami in Resources: Modular Origami

A container:

A tetrahedron made with four balls:

Volume of a tetrahedron: tetrahedron made with four balls | matematicasVisuales

In Rothenburg ob der Tauber (Germany):

Volume of a tetrahedron: tetrahedron in Rothenburg (Germany) | matematicasVisuales

Tensegrity:

Volume of a tetrahedron: Tensegrity | matematicasVisuales

You can see more examples of tensegrity in Resources: Tensegrity

More origami: octahedron and tetrahedron.

Volume of a tetrahedron: origami, octahedra inside a tetrathedron  | matematicasVisuales
Volume of a tetrahedron: origami, octahedra inside a tetrahedron | matematicasVisuales

Five tetrahedra in a dodecahedron:

Volume of a tetrahedron: Five tetrahedra inside a dodecahedron | matematicasVisuales
Volume of a tetrahedron: Five tetrahedra inside a dodecahedron | matematicasVisuales
Volume of a tetrahedron: Five tetrahedra inside a dodecahedron | matematicasVisuales
Volume of a tetrahedron: Five tetrahedra inside a dodecahedron | matematicasVisuales

REFERENCES

Hugo Steinhaus - Mathematical Snapshots - Oxford University Press - Third Edition (p. 197)
Magnus Wenninger - 'Polyhedron Models', Cambridge University Press.
Magnus Wenninger - 'Dual Models', Cambridge University Press.
Peter R. Cromwell - 'Polyhedra', Cambridge University Press, 1999.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961 (p. 87).
W.W. Rouse Ball and H.S.M. Coxeter - 'Matematical Recreations & Essays', The MacMillan Company, 1947.
Luca Pacioli - De divina proportione - (La divina proporción) Ediciones Akal, 4ª edición, 2004. Spanish translation by Juan Calatrava.
Howard Eves, mathematician and historian of Mathematics, received the George Polya Award for the article Two Surprising Theorems on Cavalieri Congruence.

MORE LINKS

Hexagonal section of a cube
We can cut in half a cube by a plane and get a section that is a regular hexagon. Using eight of this pieces we can made a truncated octahedron.
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 .
A truncated octahedron made by eight half cubes
Using eight half cubes we can make a truncated octahedron. The cube tesselate the space an so do the truncated octahedron. We can calculate the volume of a truncated octahedron.
The truncated octahedron is a space-filling polyhedron
These polyhedra pack together to fill space, forming a 3 dimensional space tessellation or tilling.
The volume of a truncated octahedron
The truncated octahedron is an Archimedean solid. It has 8 regular hexagonal faces and 6 square faces. Its volume can be calculated knowing the volume of an octahedron.
The volume of a cuboctahedron
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of a cube.
The volume of a cuboctahedron (II)
A cuboctahedron is an Archimedean solid. It can be seen as made by cutting off the corners of an octahedron.
The volume of an stellated octahedron (stella octangula)
The stellated octahedron was drawn by Leonardo for Luca Pacioli's book 'De Divina Proportione'. A hundred years later, Kepler named it stella octangula.
Resources: Building Polyhedra with cardboard (Plane Nets)
Using cardboard you can draw plane nets and build polyhedra.
Resources: How to build polyhedra using paper and rubber bands
A very simple technique to build complex and colorful polyhedra.
Resources: Building polyhedra gluing faces
Using cardboard you can build beautiful polyhedra cutting polygons and glue them toghether. This is a very simple and effective technique. You can download several templates. Then print, cut and glue: very easy!
Resources: Building polyhedra using tubes
Examples of polyhedra built using tubes.
Resources: Modular Origami
Modular Origami is a nice technique to build polyhedra.
Resources: Tensegrity
Examples of polyhedra built using tensegrity.
Resources: Building polyhedra using Zome
Examples of polyhedra built using Zome.
Leonardo da Vinci: Drawing of a truncated octahedron 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 truncated octahedron.
Leonardo da Vinci: Drawing of a cuboctahedron 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 cuboctahedron.
Leonardo da Vinci: Drawing of an stellated octahedron (stella octangula) 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 stellated octahedron (stella octangula).
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.
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.
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.
Sections on a tetrahedron
Special sections of a tetrahedron are rectangles (and even squares). We can calculate the area of these cross-sections.
Kepler: The Area of a Circle
Kepler used an intuitive infinitesimal approach to calculate the area of a circle.
Plane developments of geometric bodies (1): Nets of prisms
We study different prisms and we can see how they develop into a plane net. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (3): Cylinders
We study different cylinders and we can see how they develop into a plane. Then we explain how to calculate the lateral surface area.
Plane developments of geometric bodies (5): Pyramid and pyramidal frustrum
Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.
Plane developments of geometric bodies (7): Cone and conical frustrum
Plane developments of cones and conical frustum. How to calculate the lateral surface area.
Plane developments of geometric bodies: Dodecahedron
The first drawing of a plane net of a regular dodecahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .