Trapezoid Calculator
Results
Formulas Used in Trapezoid Calculator
In-Depth Tutorial: Trapezoid Calculator
A trapezoid (called a trapezium in UK English, and confusingly, "trapezium" in US English refers to a quadrilateral with NO parallel sides) is any four-sided figure with at least one pair of parallel sides. The parallel sides are called the bases (b₁ and b₂), and the perpendicular distance between them is the height (h). This tutorial covers the three sub-types this calculator handles — general, isosceles, and right — and walks through 3 worked examples.
The universal area formula
Area = ½ × (b₁ + b₂) × h. This works for every trapezoid, regardless of type. The intuition: average the two bases (giving a "typical width"), then multiply by the height — the trapezoid is essentially a rectangle of width (b₁+b₂)/2 and height h.
Example 1 — general trapezoid: b₁ = 8, b₂ = 4, h = 3. Area = ½ × (8 + 4) × 3 = ½ × 12 × 3 = 18. Midsegment m = (b₁ + b₂) / 2 = 6.
Isosceles trapezoid (non-parallel sides equal)
An isosceles trapezoid has the two non-parallel sides (legs) equal in length. This unlocks two extra properties: equal diagonals and equal base angles. If you only know the bases and a leg, the height is derived:
h = √(leg² − ((b₁ − b₂) / 2)²)
Derivation: drop perpendiculars from each end of b₂ down to b₁. This creates two right triangles on the sides, each with hypotenuse = leg and horizontal leg = (b₁ − b₂) / 2. Pythagorean theorem gives the vertical leg = h.
Example 2 — isosceles trapezoid: b₁ = 10, b₂ = 6, leg = 5. h = √(25 − ((10−6)/2)²) = √(25 − 4) = √21 ≈ 4.58. Area = ½ × 16 × 4.58 ≈ 36.66. Perimeter = 10 + 6 + 2×5 = 26.
Right trapezoid (two adjacent right angles)
A right trapezoid has two adjacent right angles, so one of the non-parallel sides is itself perpendicular to both bases. That perpendicular leg IS the height — no extra derivation needed. The oblique (slanted) leg follows from Pythagoras:
oblique leg = √(h² + (b₁ − b₂)²)
Example 3 — right trapezoid: b₁ = 12, b₂ = 7, perpendicular leg = h = 4. Oblique leg = √(16 + 25) = √41 ≈ 6.40. Area = ½ × 19 × 4 = 38. Perimeter = 12 + 7 + 4 + 6.40 = 29.40.
The midsegment
The midsegment connects the midpoints of the two non-parallel legs. It is always parallel to the bases, lies exactly halfway between them, and its length is the average of the bases: m = (b₁ + b₂) / 2. This is more than trivia — the midsegment is the geometric mean line that lets you compute area as m × h (instead of ½(b₁+b₂)h), a useful sanity check.
Diagonals
General trapezoid diagonals have no simple closed form (you need additional inputs like an interior angle). For the isosceles case, both diagonals are equal — a defining property used in many proofs. For a right trapezoid, the two diagonals usually differ; compute each using the right triangle they form with the bases.
Interior angles
Every trapezoid has interior angles summing to 360° (true for any quadrilateral). Additionally, the two angles on each non-parallel leg are co-interior between the parallel bases — they sum to 180°. So if you know one angle on a leg, you know the other (180° − it). For isosceles trapezoids, the two base angles on each base are equal.
Common mistakes
- Confusing US vs UK "trapezoid". US "trapezoid" = at least one pair of parallel sides. UK "trapezium" = same thing. US "trapezium" = NO parallel sides (rare term). The calculator uses the US convention.
- Mixing bases with legs. Bases are the two parallel sides; legs are the two non-parallel sides. Always identify the bases first — the area formula needs their sum, not the legs.
- Using the slanted leg as height. Height is always the perpendicular distance between bases. Only in a right trapezoid does the leg equal the height; for general / isosceles, you must compute h separately.
- Forgetting that midsegment = average of bases. Some students use m = b₁ − b₂ (no — that's the horizontal offset). Always (b₁ + b₂) / 2.
When to use a different calculator
- For just isosceles trapezoid problems, the dedicated Isosceles Trapezoid Calculator auto-fills the leg-symmetry assumption.
- For right trapezoid problems with the perpendicular leg given, the Right Trapezoid Calculator skips the type selector.
- For trapezoid angle problems (finding all 4 interior angles from given constraints), see the Trapezoid Angle Calculator.
- For trapezoids defined by 4 vertex coordinates, use the Quadrilateral Shape Calculator with the Shoelace formula.
Frequently Asked Questions – Trapezoid Calculator
Area = ½ × (b₁ + b₂) × h, where b₁ and b₂ are the two parallel bases and h is the perpendicular height between them. This works for every trapezoid — general, isosceles, or right.
If you know both bases and a leg of an isosceles trapezoid: h = √(leg² − ((b₁ − b₂)/2)²). For a right trapezoid, the perpendicular leg IS the height. For a general trapezoid, you need additional information such as a diagonal or interior angle.
Yes — the diagonals of an isosceles trapezoid are always equal in length. This is a key defining property and is often used in proofs to identify a quadrilateral as an isosceles trapezoid.
Yes. Drop perpendiculars from the endpoints of the shorter base down to the longer base. If the two right triangles formed are congruent (typically by HL or SAS — sharing the height as one leg), the two slanted legs of the trapezoid are equal, proving it is isosceles.
Yes — the four interior angles of any trapezoid sum to 360°, the same as every quadrilateral. Additionally, the two angles on each non-parallel leg are co-interior angles between the parallel bases and sum to 180°.
The midsegment connects the midpoints of the two non-parallel legs. It is always parallel to the bases, lies exactly halfway between them, and its length equals the average of the bases: m = (b₁ + b₂) / 2.