A trapezoid (called a “trapezium” in British English) is a quadrilateral with at least one pair of parallel sides. Its area formula is:
A = ½(b₁ + b₂)h
where b₁ and b₂ are the lengths of the two parallel sides (the “bases”) and h is the perpendicular distance between them (the “height”). This guide proves the formula geometrically, walks through 4 worked examples, and covers special cases including isosceles trapezoids and the British “trapezium” terminology gotcha.
A trapezoid has four sides. Exactly two of them are parallel — those are the bases, labeled b₁ (the longer one by convention) and b₂ (the shorter one).
The other two sides are the legs. They are not parallel to each other in a general trapezoid; if they were, the figure would be a parallelogram instead.
The height h is the perpendicular distance between the two parallel sides — measured straight across, not along a leg.
One quick visual proof: take two identical copies of the trapezoid, flip one upside-down, and join them along a leg. The combined figure is a parallelogram with base (b₁ + b₂) and height h. The parallelogram’s area is base × height = (b₁ + b₂) × h. Since this contains two copies of the original trapezoid, one trapezoid’s area is half: (b₁ + b₂) × h / 2.
An alternative proof: cut the trapezoid with a diagonal into two triangles. One triangle has base b₁ and height h (area = ½ × b₁ × h). The other has base b₂ and height h (area = ½ × b₂ × h). Sum: ½b₁h + ½b₂h = ½(b₁ + b₂)h.
Either way, the formula falls out cleanly.
A trapezoid has parallel sides 8 cm and 12 cm, and the perpendicular distance between them is 5 cm.
A = ½(8 + 12)(5) = ½(20)(5) = ½(100) = 50 cm².
A trapezoid has area 60 cm², bases 8 cm and 12 cm. Find its height.
60 = ½(8 + 12) × h
60 = ½ × 20 × h
60 = 10h
h = 6 cm
The formula rearranges cleanly: h = 2A / (b₁ + b₂).
An isosceles trapezoid has bases 6 and 10, and slant legs of length 5. Find its area.
An isosceles trapezoid has two legs of equal length. By symmetry, if you drop perpendiculars from the endpoints of the shorter base to the longer base, they cut off two congruent right triangles at each end. Each triangle has hypotenuse 5 (the leg) and horizontal leg (10 − 6) / 2 = 2.
By the Pythagorean theorem, the height (vertical leg) is h = √(5² − 2²) = √21 ≈ 4.58.
Area = ½(6 + 10)(√21) = 8√21 ≈ 36.66 cm².
A trapezoid has vertices at (0, 0), (6, 0), (4, 3), and (1, 3). Find its area.
The bases are the two horizontal segments (since both pairs of points have matching y-values). Top base: from (1, 3) to (4, 3) has length 3. Bottom base: from (0, 0) to (6, 0) has length 6. Height: vertical distance between y = 0 and y = 3 is h = 3.
A = ½(3 + 6)(3) = ½(9)(3) = 13.5 square units.
The area formula works for all three types — only the base lengths and the height matter.
In American English, a trapezoid has at least one pair of parallel sides, and a trapezium is a quadrilateral with NO parallel sides.
In British English (and most of the world outside the US), the meanings are reversed: a trapezium has at least one pair of parallel sides, and a trapezoid has none.
If you are reading a textbook or paper, check which convention it uses. The formula A = ½(b₁ + b₂)h applies to the shape with parallel sides — whichever name the source uses for it.
Most modern American textbooks use the inclusive definition: a trapezoid has at least one pair of parallel sides. Under this definition, parallelograms are special cases of trapezoids (they have two pairs).
The older exclusive definition required exactly one pair of parallel sides, ruling out parallelograms. Most contemporary geometry uses the inclusive definition because it makes theorems and formulas more general.
For trapezoid area problems with all four sides given, use the Trapezoid Calculator. For the general “area of any quadrilateral” case where the figure isn’t necessarily a trapezoid, see Area of Any Polygon. For the related parallelogram (a trapezoid with both pairs of sides parallel), see the Parallelogram Calculators.
How do I find the area of a trapezoid given only its four sides? You need either the height or enough information to derive it. With four sides given but no perpendicular height, you can use the diagonal lengths or solve via the Pythagorean theorem if the trapezoid is isosceles or has a right angle. For arbitrary trapezoids from four side lengths alone, the area is not uniquely determined.
What’s the difference between a trapezoid and a parallelogram? A parallelogram has both pairs of opposite sides parallel. A trapezoid (American) has at least one pair parallel. Under the inclusive definition, every parallelogram is a trapezoid; under the exclusive definition, no parallelogram is a trapezoid.
What’s the area of a regular trapezoid? “Regular trapezoid” is not a standard term — most trapezoids are not regular in the polygon sense (only regular polygons have all sides AND angles equal, which would make them parallelograms or worse, squares). You may have heard “isosceles trapezoid” — see the section above.