Rhombus

Properties

• All four sides are equal.
• Opposite angles are equal.
• The diagonals cut each other in half at right angles.
• Each diagonal is a line of symmetry.
• A square is a rhombus with right angles.

Why the formulas work

The two diagonals cut the rhombus into four equal right-angled triangles, each with legs p/2 and q/2. Four of them have a total area of 4 × ½ × p/2 × q/2, which is half of p times q. The side of the rhombus is the hypotenuse of one of those triangles.

Worked example

Take a rhombus with diagonal p 16 cm and diagonal q 12 cm.

1. Area. Area = ½ × p × q. Here: ½ × 16 × 12 = 96 cm².
2. Side. Side = ½ √(p² + q²). Here: ½ × √(16² + 12²) = 10 cm.
3. Perimeter. Perimeter = 4 × side. Here: 4 × 10 = 40 cm.

Finding a missing measurement

If the side and one diagonal are known, use the right-angled triangle made by the half diagonals. A side of 10 cm and a diagonal of 16 cm give half of the other diagonal as √(100 − 64) = 6 cm, so that diagonal is 12 cm and the area is ½ × 16 × 12 = 96 cm². If the area and one diagonal are known, the other is 2 × area ÷ the known diagonal.

Mistakes to avoid

• Multiplying the diagonals without halving.
• Using the full diagonals in Pythagoras for the side. Use half of each.
• If the base and height are given, the area is simply base times height, as for any parallelogram.

Where you see it

The diamond suit on cards, many kites, tile patterns, and the meshes of a chain-link fence are rhombuses.

Try these

Work each one out, then tap to check.

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