Solid of revolution


The solid of revolution is a geometric body that can be formed by rotating a plane surface around a line called the axis.

A solid of revolution is, from another perspective, a three-dimensional figure that is characterized because its surface is not flat, but is curved.

It should be noted that solids of revolution can take different shapes, even irregular ones, like the one we see in the image below.

Another point to take into account is that the flat surface that rotates to form the solid may or may not intersect with the axis of revolution, as in the case of the figure called torus, which we will see later.

From the mathematical point of view, if we have two functions, we will obtain a solid of revolution if we rotate the plane region contained between these functions around a given line, which would be the axis of revolution.

It should also be noted that the axis of revolution can be not only a straight line, but also the X Axis or the Y Axis of the Cartesian plane.

Principal solids of revolution

The main solids of revolution are the following:

  • Cone: The cone is a solid of revolution that is generated by rotating a right triangle around one of its legs.
  • Cylinder: The cylinder is defined as that solid that is formed by rotating a rectangle around an axis.
  • Sphere: The sphere is a solid that is obtained by rotating a semicircle around an axis.
  • Toroid: It is the solid that is formed by rotating a polygon or a curve around the axis, leaving a hollow or empty space in the center, as we see in the figure below. When the turning curve is closed, the figure is called a torus, as we see in the image below.

Volume of a solid of revolution

In general, to calculate the volume of a solid of revolution, integral calculus can be used. One way, called the disc method, consists of dividing the figure into infinite discs or circular portions, adding together their volumes.

Another method is that of layers, used when we have a hollow figure like the torus, where the axis of revolution is not contained in the plane region that rotates. In this case, the dimension of the layer has to be calculated, which can be a parallelepiped (polyhedron with six faces that are all parallelograms), which is wrapped or rolled to generate the solid.

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