160. The Rhombus Area Secret · half the product of diagonals

Try to break it

Drag A along one diagonal and B along the other. The diagonals d₁ and d₂ change length, but the area always equals ½ × d₁ × d₂. Try to find a setting where the area doesn't match the formula — impossible.

How you build it

Construct a rhombus from two diagonals.

• Mark point A on the canvas.
• Mark point C to the right of A.
• Draw segment AC — this is the first diagonal of the rhombus.
• Construct the perpendicular bisector of AC. It passes through the midpoint O.
• Mark O where the bisector crosses AC — the centre of the rhombus.
• With centre O, draw an arc cutting the bisector above AC.
• Mark B where the arc meets the bisector above AC.
• Mark D where the arc meets the bisector below AC.
• Draw segment AB, one side of the rhombus.
• Draw segment BC.
• Draw segment CD.
• Draw segment DA to close the rhombus.

The proof, step by step

Prove that the area of a rhombus is half the product of its diagonals.

• The diagonals of a rhombus bisect each other at right angles.
• The rhombus is divided into 4 congruent right-angled triangles.
• Area of one triangle = 1/2 × (d1/2) × (d2/2).
• Total Area = 4 × 1/2 × (d1/2) × (d2/2) = 1/2 × d1 × d2.

Worked example

The diagonals of a rhombus are 18 cm and 24 cm. Find its area.

Area = 1/2 × d1 × d2 = 1/2 × 18 × 24 = 216 cm².

• 216 cm² — correct
• 432 cm²
• 108 cm²
• 864 cm²