(i)  For a cylinder, there are infinitely many of planes of symmetry.
     They constitute a continuous set of planes.
     They all are through the axis of rotation.

     There is one axis of rotation.



(ii)  For a cone, there are infinitely many of planes of symmetry.
      They constitute a continuous set of planes.
      They all are through the axis of rotation.

      There is one axis of rotation.


(iii) For a square based pyramid, there are 4 (four) planes of symmetry.
      They all are through the altitude to the base.
      Two of the four planes of symmetry are through diagonals of the square at the base.
      Two other of the four planes of symmetry are through midpoints 
      of the opposite sides of the square at the base.

      There is one axis of rotation, but the rotations do not constitute a continuous set
      of rotations. There are 4 (four) discrete rotations around the rotation axis 
      by the angles 90°, 180°, 270° and 360°.



(iv)  For a rectangle based pyramid, the rectangle at the base is assumed not to be a square.
      There are 2 (two) planes of symmetry.
      They all are through the altitude to the base.
      These two planes of symmetry are through midpoints of the opposite sides 
      of the rectangle at the base.

      There is one axis of rotation, but the rotations do not constitute a continuous set
      of rotations. There are only 2 (two) discrete rotations around the rotation axis 
      by the angles 180° and 360°.



(v)   For an equilateral triangle based pyramid, there are 3 (three) planes of symmetry.
      They all are through the altitude to the base.
      Each of these three planes of symmetry are through one vertex at the base and the midpoint 
      of the opposite side of the equilateral triangle  at the base.

      There is one axis of rotation, but the rotations do not constitute a continuous set
      of rotations. There are only 3 (three) discrete rotations around the rotation axis 
      by the angles 120°, 240° and 360°.



(vi)  For an isosceles triangle based pyramid, the isosceles triangle at the base is assumed 
      not to be equilateral.
      There is  1 (one) planes of symmetry.
      It is through the altitude to the base.
      This unique plane of symmetry is through the vertex of the triangle at the base 
      and the midpoint of the opposite side of the isosceles triangle  at the base.

      There is no axis of rotation.



(vii) For a scalene triangle based pyramid, the scalene triangle at the base is assumed 
      not to be isosceles.
      There are no planes of symmetry.

      There is no axis of rotation.