Rechtwinkliges-Trapez-Rechner

A right trapezoid is a trapezoid with two adjacent right angles — meaning one of its non-parallel sides (the legs) is perpendicular to both parallel bases. This perpendicular leg acts as the height of the trapezoid, simplifying many calculations. The other leg is the "slanted" or "oblique" leg, and its length follows from the Pythagorean theorem. This tutorial covers the formulas, three worked examples, and how right trapezoids appear in real-world structures.

The setup

A right trapezoid ABCD has:

• Two parallel bases: AB (longer, b₁) and CD (shorter, b₂)
• One leg perpendicular to both bases (call it AD = h, the height)
• One slanted leg connecting the other two vertices (call it BC = ℓ, the oblique leg)
• Two right angles at the vertices where the perpendicular leg meets each base

The right trapezoid is essentially "half of an isosceles trapezoid cut down the middle plus a rectangle" — but more commonly seen as a wedge or a ramp profile.

Area formula

The area is the same as any trapezoid:

A = ½ × (b₁ + b₂) × h

The simplification for right trapezoids: the height h is just the perpendicular leg AD — no need to compute it separately. You can read it directly from the right-angle leg.

Oblique leg formula

The slanted leg BC connects the end of the shorter base to the end of the longer base. By the Pythagorean theorem applied to the right triangle formed by the height (h) and the horizontal base difference (b₁ − b₂):

ℓ = √(h² + (b₁ − b₂)²)

Where this comes from: drop a perpendicular from C (end of shorter base) to AB at point E. The segment CE = h (the height). The segment EB = b₁ − b₂ (the horizontal offset). The slanted leg BC is the hypotenuse of right triangle CEB: ℓ² = h² + (b₁−b₂)².

The angle of the slanted leg

The slanted leg makes an angle θ with the longer base. From the right triangle CEB:

tan(θ) = h / (b₁ − b₂)

So θ = arctan(h / (b₁ − b₂)).

Equivalently: θ is the angle of inclination — important for ramps, roofs, and wedges.

Worked example 1 — basic right trapezoid

Right trapezoid with b₁ = 10, b₂ = 6, h = 4.

Area = ½ × (10 + 6) × 4 = ½ × 16 × 4 = 32.

Slanted leg ℓ = √(4² + (10−6)²) = √(16 + 16) = √32 = 4√2 ≈ 5.66.

Perimeter = 10 + 6 + 4 + 4√2 ≈ 25.66.

Angle of slanted leg: tan(θ) = 4/4 = 1, so θ = 45°.

Worked example 2 — find height from area

Right trapezoid with bases 12 and 8, area 50. Find the height.

From A = ½(b₁ + b₂) × h: 50 = ½ × 20 × h = 10h → h = 5.

Worked example 3 — engineering — ramp profile

A construction ramp is 4 m long horizontally at the base, 2 m at the top, and rises 1.5 m. Compute the slanted-leg length and ramp angle.

ℓ = √(1.5² + (4−2)²) = √(2.25 + 4) = √6.25 = 2.5 m.

Angle of inclination: tan(θ) = 1.5 / 2 = 0.75 → θ = arctan(0.75) ≈ 36.87°.

Right trapezoid as half of a different shape

A right trapezoid can be visualized as:

• A rectangle minus a right triangle. Cut a right triangle off one corner of a rectangle to make a right trapezoid.
• A rectangle plus a right triangle. Attach a right triangle to one side of a rectangle.
• Half of an isosceles trapezoid. Cut an isosceles trapezoid down its axis of symmetry.

Each of these decompositions gives an alternative way to compute the area (sometimes easier than the direct formula).

Perimeter

Perimeter = b₁ + b₂ + h + ℓ.

The four sides: two parallel bases, one perpendicular leg (= height), one slanted leg. All four must be included.

Diagonals

The two diagonals of a right trapezoid have different lengths in general:

Diagonal from A to C (across the rectangle-like portion): d₁ = √(b₂² + h²)
Diagonal from B to D (across the wedge): d₂ = √(b₁² + h²)

The longer diagonal is the one spanning the longer base.

Real-world applications

• Construction ramps. Wheelchair ramps, loading docks, and ramp profiles for vehicles.
• Roof profiles. The side view of a "shed roof" (single-sloped) is a right trapezoid.
• Wedge-shaped lots. Real-estate parcels with one straight road frontage and one diagonal boundary form right trapezoids.
• Engineering supports. Brackets and supports often have right-trapezoid profiles.
• Geometry of building cuts. Stair stringers and roof rafters often involve right trapezoid sections.

Right trapezoid vs general trapezoid

Common mistakes

• Mistaking the slanted leg for the height. Height is the PERPENDICULAR leg. The oblique leg is longer.
• Computing the slanted leg with the wrong base. The horizontal offset is (b₁ − b₂), not b₁ or b₂ alone.
• Forgetting one of the four sides in perimeter. A trapezoid has 4 sides — perimeter includes all of them.
• Treating a right trapezoid as having TWO perpendicular legs. Only ONE leg is perpendicular. The other is slanted.