Here are the differences between a square and a rhombus.
Square. Its properties are
(a) All sides are equal.
(b) Opposite sides are equal and parallel.
(c) All angles are equal to 90 degrees.
(d) The diagonals are equal.
(e) Diagonals bisect each other at right angles.
(f) Diagonals bisect the angles.
(g) The intersection of the diagonals is the circumcentre. That is, you can draw a circle with that as centre to pass through the four corners.
(h) The intersection of the diagonals is also the incentre. That is, you can draw a circle with that as centre to touch all the four sides.
(i) Any two adjacent angles add up to 180 degrees.
(j) Each diagonal divides the square into two congruent isosceles right-angled triangles.
(k) The sum of the four exterior angles is 4 right angles.
(l) The sum of the four interior angles is 4 right angles.
(m) Lines joining the mid points of the sides of a square in an order form another square of area half that of the original square.
(n) If through the point of intersection of the two diagonals you draw lines parallel to the sides, you get 4 congruent squares each of whose area will be one-fourth that of the original square.
(o) Join the quarter points of a diagonal to the vertices on either side of the diagonal and you get a rhombus of half the area of the original square.
(p) Revolve a square about one side as the axis of rotation and you get a cylinder whose diameter is twice the height.
(q) Revolve a square about a line joining the midpoints of opposite sides as the axis of rotation and you get a cylinder whose diameter is the same as the height.
(r) Revolve a square about a diagonal as the axis of rotation and you get a double cone attached to the base whose maximum diameter is the same as the height of the double cone.
Rhombus. Its properties are
(a) All sides are equal.
(b) Opposite sides are parallel.
(c) Opposite angles are equal.
(d) Diagonals bisect each other at right angles.
(e) Diagonals bisect the angles.
(f) Any two adjacent angles add up to 180 degrees.
(g) The sum of the four exterior angles is 4 right angles.
(h) The sum of the four interior angles is 4 right angles.
(i) The two diagonals form four congruent right angled triangles.
(j) Join the mid-points of the sides in order and you get a rectangle.
(k) Join the mid-points of the half the diagonals in order and you get a rhombus.
(l) The distance of the point of intersection of the two diagonals to the mid point of the sides will be the radius of the circumscribing of each of the 4 right-angled triangles.
(m) The area of the rhombus is a product of the lengths of the 2 diagonals divided by 2.
(n) The lines joining the midpoints of the 4 sides in order, will form a rectangle whose length and width will be half that of the main diagonals. The area of this rectangle will be one-half that of the rhombus.
(o) If through the point of intersection of the two diagonals you draw lines parallel to the sides, you get 4 congruent rhombus each of whose area will be one-fourth that of the original rhombus.
(p) There can be no circumscribing circle around a rhombus.
(q) There can be no inscribed circle within a rhombus.
(r) Two congruent equilateral triangles are formed if the shorter diagonal is equal to one of the sides.
(s) Two congruent isosceles acute triangles are formed when cut along the shorter diagonal.
(t) Two congruent isosceles obtuse triangles are formed when cut along the longer diagonal.
(u) Four congruent RATs are formed when cut along both the diagonals. These RATs cannot be isosceles RATs.
(v) Join the quarter points of both the diagonals and you get a similar rhombus of 1/4th area as the parent rhombus.
(w) Revolve a rhombus about any side as the axis of rotation and you get a cylindrical surface with a convex cone at one end a concave cone at the other end. Their slant heights will be the same as the cylindrical sides of the solid.
(x) Revolve a rhombus about a line joining the midpoints of opposite sides as the axis of rotation and you get a cylindrical surface with concave cones at the both ends.
(y) Revolve a rhombus about the longer diagonal as the axis of rotation and you get a solid with two cones attached at their bases. The maximum diameter of the solid will be the same as the shorter diagonal of the rhombus.
(z) Revolve a rhombus about the shorter diagonal as the axis of rotation and you get a solid with two cones attached at their bases. The maximum diameter of the solid will be the same as the longer diagonal of the rhombus.