matematicas visuales visual math

The solid cut from an infinite circular cylinder by two planes is a cylindrical segment or a truncated cylinder. The simplest case is when one of the cutting planes is perpendicular to the axis of the cylinder. Then the cylindrical segment has a circular base.

The main interest of this page is to see how a truncated cylinder can be developed into a plane.

Truncated cylinder or cylindrical segment: a truncated cylinder| matematicasVisuales
Truncated cylinder or cylindrical segment: a truncated cylinder developing| matematicasVisuales
Truncated cylinder or cylindrical segment: plane development of a truncated cylinder | matematicasVisuales

This in another example:

Truncated cylinder or cylindrical segment: plane development of a truncated cylinder | matematicasVisuales

The volume of a cylindrical segment it is easy to obtain if we notice that two copies of the cylindrical segment one of them turned upside-down, together form a cylinder.

Truncated cylinder or cylindrical segment: volume | matematicasVisuales


The section of a cylinder by an oblique plane looks like an ellipse. This is true and a beautiful demonstration uses Dandelin Spheres.

Ellipses as sections of cylinders: Dandelin Spheres
The section of a cylinder by a plane cutting its axis at a single point is an ellipse. A beautiful demonstration uses Dandelin Spheres.


REFERENCES

Hilbert and Cohn-Vossen. Geometry and the Imagination. Chelsea Publishing Company. pag.7.

MORE LINKS

Ellipses as sections of cylinders: Dandelin Spheres
The section of a cylinder by a plane cutting its axis at a single point is an ellipse. A beautiful demonstration uses Dandelin Spheres.
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.
Albert Durer and ellipses: cone sections.
Durer was the first who published in german a method to draw ellipses as cone sections.
Albert Durer and ellipses: Symmetry of ellipses.
Durer made a mistake when he explanined how to draw ellipses. We can prove, using only basic properties, that the ellipse has not an egg shape .
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.
Ellipse and its foci
Every ellipse has two foci and if we add the distance between a point on the ellipse and these two foci we get a constant.
Equation of an ellipse
Transforming a circle we can get an ellipse (as Archimedes did to calculate its area). From the equation of a circle we can deduce the equation of an ellipse.
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.
Cavalieri: The volume of a sphere
Using Cavalieri's Principle we can calculate the volume of a sphere.
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.