Rhombus

01 What is always true

∠DAB means the angle at A — the middle letter is always the corner, and the outer two say which sides form it.

• All four sides are the same length.

  AB = BC = CD = DA, each 60 units here.

  If a square has four equal sides, is a square a rhombus?

  1. Yes. Read the definition again: a rhombus is a parallelogram with four equal sides. That is the whole test.

  2. A square has four equal sides. Here it is with the same four ticks a rhombus carries.

  3. So it passes the test, and a square is a rhombus. Its right angles are something extra it has as well — nothing in the definition says a rhombus may not have them.

  4. The one that does not work the other way round: most rhombuses are not squares, because their angles are not 90°. Every square is a rhombus; only some rhombuses are squares.

• 2 pairs of sides are parallel.

  AB is parallel to CD, BC is parallel to DA

• Opposite angles are equal.

  ∠DAB = ∠BCD (60°), ∠ABC = ∠CDA (120°)

• Any two angles next to each other add up to 180°.

  ∠DAB + ∠ABC = 60° + 120° = 180°, ∠ABC + ∠BCD = 120° + 60° = 180°

• All 4 angles add up to 360°.

  True of every quadrilateral, however odd it looks.

• The diagonals cross at right angles and cut each other in half.

  AC = 103.9 and BD = 60, meeting at 90° at the midpoint of both.

02 Why it works

A rhombus is a parallelogram first. Everything a parallelogram has, it has: opposite sides parallel, opposite angles equal, neighbouring angles adding to 180°. The four equal sides are the extra, not a replacement.

The property that is only a rhombus's is the last row: its diagonals cross at right angles. A parallelogram's diagonals bisect each other but meet at a slant; make all four sides equal and they square up.

It is often drawn balanced on a corner, which is why it gets called a diamond. Turning a shape never changes what it is — the same rhombus set flat on a side is still a rhombus.

03 Where it sits

See the whole family on Quadrilaterals.