In Plane developments of geometric bodies (1): Nets of prisms we can see how regular and non-regular prisms can be developed into a plane. Now we are going to study nets of prisms cut by an oblique plane.

This is one example:

In the examples above a base was a regular polygons. But we can consider prisms whose bases are not regular polygons. In the next mathlet, bases are non-regular polygons (although they are inscribed in a circle and they are convex polygons). Each time we change the number of sides of the base a new prism is generated with sides randomly drawn:

This is one example of a non-regular transparent prism cut by an oblique plane:

Two examples of nets of non-regular prisms cut by an oblique plane:

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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 (5): Pyramid and pyramidal frustrum Plane net of pyramids and pyramidal frustrum. How to calculate the lateral surface area.

Plane developments of geometric bodies (6): Pyramids cut by an oblique plane Plane net of pyramids cut by an oblique plane.

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 (8): Cones cut by an oblique plane Plane developments of cones cut by an oblique plane. The section is an ellipse.

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.

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.

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.

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.