What happens if a flat geometric shape, such as a rectangle, begins to rotate rapidly relative to one of its sides? We create a new geometric solid in space by rotation.
The sides of a rotating rectangle form the side surfaces of the cylinder.
The official definition for such geometric solids is as follows:
The solids of revolution are volume solids that arise when a flat geometric figure bounded by a curve rotates around an axis lying in the same plane.
Here it is important that a flat geometric figure can be a completely arbitrary shape.
For example, a curve that, when rotated, forms a vase or a light bulb. Such tools for creating bodies of revolution are very popular with those who work in 3D-design programs.
But from a mathematical point of view, the following geometric solids of revolution are of primary interest to us:
The cylinder is formed by a rectangle rotating around one of the sides.
The coneis formed by a rectangular triangle rotating around one of the sides.
A truncated cone is part of the cone located between its base and the cutting plane parallel to the base.
It is formed by rotating a rectangular trapezoid around its side, perpendicular to the trapezoid bases.
The ball is formed by a semicircle rotating around the diameter of the cut. During the rotation of the figures' contours, a surface of revolution arises (for example, a sphere formed by a circle). During the rotation of filled contours, bodies appear (like a ball formed by a circle).
An ellipsoid is a surface in three-dimensional space, obtained by deforming a sphere along three mutually perpendicular axes.
The torus is formed by a circle rotating around a straight line that does not intersect it. The usual sense of the torus is a "bagel".
A paraboloid is a surface formed due to a rotation around the axis of a curve formed by a parabola graph. Hence the name paraboloid.
A hyperboloid is a surface formed due to rotation around the axis of a curve formed by a hyperbola graph. Accordingly, the name is a hyperboloid.