195. The Rhombus Area Secret · Diagonals hold the key to the area

Try to break it

Drag A along the vertical diagonal to change d₁, and B along the horizontal diagonal to change d₂. The two diagonals stay perpendicular and bisect each other, and the area always equals ½ × d₁ × d₂. Try to find a setting where it disagrees with the formula; impossible.

How you build it

Construct a rhombus from its diagonals.

• Pick the Segment tool and draw diagonal AC. This is the first diagonal of the rhombus.
• Click on segment AC with the Bisector tool. The perpendicular bisector passes through the midpoint O — this is the line on which B and D will lie.
• Pick the Arc tool. Click A as the centre and drag to draw an arc that crosses the perpendicular bisector on both sides. The two crossing points are B and D. Any point on the perpendicular bisector of AC is equidistant from A and C — so AB = CB and AD = CD, making all four sides equal.
• Connect vertex A to vertex B.
• Connect vertex B to vertex C.
• Connect vertex C to vertex D.
• Connect vertex D back to vertex A to complete the rhombus.

The proof, step by step

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

• In a rhombus, the diagonals bisect each other at 90°. Let the diagonals be AC and BD intersecting at O.
• The area of the rhombus is the sum of the areas of two triangles: Area(ABCD) = Area(ΔABC) + Area(ΔADC).
• Area(ΔABC) = ½ × base × height = ½ × AC × OB.
• Area(ΔADC) = ½ × base × height = ½ × AC × OD.
• Total Area = ½ × AC × OB + ½ × AC × OD = ½ × AC × (OB + OD). Since OB + OD = BD, Area = ½ × AC × BD.

Worked example

The diagonals of a rhombus measure 16 cm and 12 cm. What is the area of the rhombus?

Area = ½ × d1 × d2 = ½ × 16 × 12 = 96 cm².

• 96 cm² — correct
• 192 cm²
• 48 cm²
• 120 cm²