Trapezium Calculator

Instant, accurate calculations for area, perimeter, diagonals, legs, angles, and median. Interactive diagram updates as you type.

For a general trapezium: one pair of parallel sides (bases). The legs may be unequal. The calculator automatically computes height, diagonals, and angles from the four side lengths.

Conditions: |b-a| < c+d and |c-d| < |b-a| (triangle inequality for the two triangles formed by height).

Non‑isosceles (5,9,4,5) Isosceles (4,7,3,3) 6,10,4,5 3,8,3.5,4.2 2.5,5.5,2.8,3.1
Properties
Area
25.46
sq units
Perimeter
23.00
units
Height
3.92
units
Median
7.00
units
Diagonal AC
7.16
units
Diagonal BD
7.81
units
Left base angle
73.0
°
Right base angle
60.3
°

Angles are the lower base angles (between bottom base and each leg).

Understanding the General Trapezium

A trapezium (or trapezoid) is a quadrilateral with at least one pair of parallel sides (the bases). Unlike an isosceles trapezium, the legs (non‑parallel sides) can have different lengths, and base angles are not necessarily equal.

Key Properties

  • Bases parallel: a ∥ b.
  • Legs may be unequal: c ≠ d possible.
  • Adjacent angles supplementary: Angles on the same side of a leg sum to 180°.
  • Midsegment (median): Parallel to bases, length = (a + b)/2.
  • Diagonals are generally not equal.
  • Height (h): Perpendicular distance between bases, computed from side lengths.

Important Formulas

Quantity Formula
Area (A) A = ½ (a + b) · h
Perimeter (P) P = a + b + c + d
Median (m) m = (a + b)/2
Height (h) h = √(c² − x₁²) where x₁ = (c² − d² + Δ²)/(2Δ), Δ = |b−a|
Diagonal d₁ (AC) √((x₁ + a)² + h²)
Diagonal d₂ (BD) √((b − x₁ − a)² + h²)
Base angle (left) θ_left = arccos((Δ² + c² − d²)/(2·c·Δ))

? Validity conditions

  • The four sides must satisfy the triangle inequalities in the two right‑angled triangles formed by the height: |b−a| < c + d and |c−d| < |b−a|.
  • If a = b (parallel sides equal), then the trapezium becomes a parallelogram; in that case c must equal d to form a valid trapezium.
  • All side lengths must be positive.

Real‑World Applications

  • Architecture: Asymmetrical trapezoidal windows, roof sections.
  • Engineering: Cross‑sections of channels with different side slopes.
  • Land surveying: Irregular plots with one pair of parallel boundaries.
  • Design: Furniture with non‑symmetrical trapezoidal shapes.
  • Mathematics: Numerical integration (trapezoidal rule) works for any trapezium.

Frequently Asked Questions

Simply enter equal leg lengths (c = d). The calculator will detect symmetry and still work correctly. Example preset "Isosceles" shows this.

The entered side lengths do not satisfy the geometric constraints for a trapezium with parallel bases. Check that |b−a| is less than c + d and greater than |c−d|.

From the law of cosines in the two triangles formed by dropping perpendiculars from the top base ends. We solve for the horizontal offset x₁ and then h = √(c² − x₁²).

Yes. A right trapezium has one leg perpendicular to the bases. You can simulate it by ensuring that one leg's horizontal projection is zero (i.e., that leg equals the height). However, our calculator works for any shape that meets the validity conditions.