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En plane developments of geometric bodies (5): Pyramid and pyramidal frustrum we can study pyramids, pyramids cut by a plane parallel to the base and their developments into a plane net. In this page we are going to see how a pyramid cut by an oblique plane can be developed into a plane net.

We have already seen similar cases in plane developments of geometric bodies (2): Prisms cut by an oblique plane and in plane developments of geometric bodies (4): Cylinders cut by an oblique plane.

Pyramids cut by an oblique plane and their nets | matematicasVisuales
Pyramids cut by an oblique plane and their nets: developing into a plane net | matematicasVisuales
Pyramids cut by an oblique plane and their nets: a plane net | matematicasVisuales

This in another example:

Pyramids cut by an oblique plane and their nets: another example of pyramid  | matematicasVisuales
Pyramids cut by an oblique plane and their nets: another example of a plane net  | matematicasVisuales

When the pyramid has a lot of faces it is, in some way, like a cone and the oblique section resembles an ellipse:

Pyramids cut by an oblique plane and their nets: with a lot of faces  | matematicasVisuales

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Plane developments of geometric bodies (8): Cones cut by an oblique plane
Plane developments of cones cut by an oblique plane. The section is an ellipse.
Plane developments of geometric bodies (4): Cylinders cut by an oblique plane
We study different cylinders cut by an oblique plane. The section that we get is an ellipse.
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 (2): Prisms cut by an oblique plane
Plane nets of prisms with a regular base with different side number cut by an oblique plane.
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: 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 .
Plane developments of geometric bodies: Octahedron
The first drawing of a plane net of a regular octahedron was published by Dürer in his book 'Underweysung der Messung' ('Four Books of Measurement'), published in 1525 .
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 .
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.
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 the tetrahedron
The volume of a tetrahedron is one third of the prism that contains it.
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 .
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.
Chamfered Cube
You can chamfer a cube and then you get a polyhedron similar (but not equal) to a truncated octahedron. You can get also a rhombic dodecahedron.
Resources: How to build polyhedra using paper and rubber bands
A very simple technique to build complex and colorful polyhedra.