Area of a Trapezoid Using Diagonals

The General Diagonal Area Formula

For any convex quadrilateral, including every standard trapezoid, the area A is given by half the product of its diagonals multiplied by the sine of the angle formed at their intersection:

Key property: Because sin(θ) = sin(180° - θ), it does not matter whether you measure the acute angle or the supplementary obtuse angle where the two diagonals cross. Both yield the exact same positive area!

Special Case: Orthodiagonal Trapezoid (Perpendicular Diagonals)

When the diagonals of a trapezoid intersect at a strict right angle (θ = 90° or π/2 radians), the sine term simplifies completely because sin(90°) = 1.

This applies to all orthodiagonal quadrilaterals (including rhombuses, kites, and square trapezoids).

Geometric Step-by-Step Proof

Why does ½ · d₁ · d₂ · sin(θ) calculate the complete trapezoid?

1. Let diagonals AC = d₁ and BD = d₂ intersect at central point P.
2. Point P partitions diagonal d₁ into lengths x₁ + x₂ = d₁, and diagonal d₂ into lengths y₁ + y₂ = d₂.
3. This divides the trapezoid into four non-overlapping triangles meeting at P:
   • Area(ΔAPB) = ½ · x₁ · y₁ · sin(θ)
   • Area(ΔBPC) = ½ · x₁ · y₂ · sin(180° - θ) = ½ · x₁ · y₂ · sin(θ)
   • Area(ΔCPD) = ½ · x₂ · y₂ · sin(θ)
   • Area(ΔDPA) = ½ · x₂ · y₁ · sin(180° - θ) = ½ · x₂ · y₁ · sin(θ)
4. Summing all four sub-triangle areas together:
   Total Area = ½ sin(θ) · [x₁y₁ + x₁y₂ + x₂y₂ + x₂y₁]
5. Factoring out common terms:
   Total Area = ½ sin(θ) · [(x₁ + x₂)(y₁ + y₂)] = ½ · d₁ · d₂ · sin(θ). (Q.E.D.)

Worked Example