Rhombus: Four Equal Sides and Diagonals That Cross at Right Angles

What has to be true before it counts

Only one condition is imposed: all four sides the same length. Everything else on the list below is a consequence of that, not an extra requirement, which is why a rhombus can be drawn leaning heavily to one side and still be a rhombus.

Note what is absent. Nothing here says the angles are 90 degrees, and nothing says the diagonals are the same length. In a typical rhombus one diagonal is noticeably longer than the other.

Diagrams mark a rhombus with a single dash on all four sides. If instead one pair carries a single dash and the other pair a double dash, only opposite sides have been declared equal and the shape is a parallelogram, where the diagonals cannot be assumed perpendicular.

• Four sides of equal length
• Opposite sides parallel, so opposite angles are equal and adjacent angles add to 180 degrees
• Diagonals bisect each other at right angles
• Each diagonal bisects the two corner angles it passes through
• Two lines of symmetry, both running along the diagonals
• Rotational symmetry of order 2

Is a square a rhombus?

Yes, and the answer is not a technicality. A square has four equal sides, which is the whole definition, so every square is a rhombus. What a square adds is four right angles, and that extra condition also forces its diagonals to be equal.

The reverse is false. A rhombus is only a square when its angles happen to be 90 degrees, so a question that gives you a rhombus has not given you right angles and you cannot use Pythagoras on the sides as though it had.

Two routes to the area, and when each is quicker

Because a rhombus is a parallelogram, base times perpendicular height works. If a diagram marks a base and a height at right angles to it, that is the shorter route.

When the diagonals are the ones labelled, use half their product instead. The right-angled crossing splits the rhombus into four congruent right-angled triangles, and adding those four areas gives exactly half of one diagonal multiplied by the other.