A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases, and the other two sides are the legs. Its area is the average of the bases times the height: A = b1 + b22 × h. In British English it is called a trapezium.
Average the two bases, then multiply by the height: 10 + 62 × 4 = 32.
Formal definition and notation
Under the exclusive (usual US school) definition, a trapezoid is a quadrilateral with exactly one pair of parallel sides, the bases b1 and b2. Its heighth is the perpendicular distance between the bases, and its area is 12(b1 + b2)h. Some texts use the inclusive definition, at least one pair of parallel sides, which makes every parallelogram a trapezoid.
Symbol
Read it as
Meaning
b1, b2
b one, b two
the two parallel sides (bases)
h
h
the height between the bases
matching arrowheads
parallel marks
show which two sides are parallel
Key facts
Exactly one pair of opposite sides is parallel (US school definition).
The two angles along each leg add to 180°, because the bases are parallel.
An isosceles trapezoid has equal legs, equal base angles and equal diagonals.
A right trapezoid has two right angles, so one leg is also the height.
The midsegment joins the midpoints of the legs; it is parallel to the bases and its length is their average.
Trapezoid area formula
Trapezoid area formula
A = b1 + b22 × h
b1, b2
the lengths of the two parallel sides
h
the perpendicular distance between them (not a slanted leg)
b1 + b22
the average base, which is also the length of the midsegment
Two copies of a trapezoid, one flipped, fit together into a parallelogram with base b₁ + b₂ and the same height. The trapezoid is half of it.
Trapezoid examples, step by step
Example 1: Area of a trapezoid
Warm-up
A trapezoid has bases of 8 m and 12 m and a height of 5 m. Find its area.
A parallelogram has two pairs of parallel sides and area bh. A trapezoid has one pair and area b1 + b22h.
Trapezoidvs.Trapezium
In British English a trapezium is what Americans call a trapezoid. Same shape, two regional names.
Quick check: 3 questions
Question 1 of 3
A trapezoid has bases of 5 ft and 9 ft and a height of 6 ft. What is its area?
Show answer and explanation
Answer: B.5 + 92 × 6 = 7 × 6 = 42. A forgets to halve, C multiplies all three numbers, and D adds them.
Question 2 of 3
Under the usual US definition, how many pairs of parallel sides does a trapezoid have?
Show answer and explanation
Answer: B. A trapezoid has exactly one pair of parallel sides, its bases. Two pairs would make a parallelogram, and a quadrilateral with no parallel sides is neither.
Question 3 of 3
In a trapezoid, one angle next to the longer base is 70°. What is the angle at the other end of that same leg?
Show answer and explanation
Answer: C. The two angles along a leg sit between parallel bases, so they add to 180°: 180 − 70 = 110. A assumes they are equal, B uses 90, and D uses 360.
Trapezoid: frequently asked questions
Is a trapezoid a parallelogram?
No. A parallelogram needs two pairs of parallel sides, and a trapezoid has only one. Under the usual US school definition (exactly one pair), the two groups do not overlap at all. Under the inclusive definition (at least one pair), every parallelogram is a trapezoid, but a trapezoid is still not a parallelogram.
Why do you divide by 2 in the trapezoid area formula?
Put two identical trapezoids together, one turned upside down, and they form a parallelogram. Its base is b1 + b2 and its height is h, so its area is (b1 + b2)h. Each trapezoid is half of that parallelogram, so you divide by 2. In effect you multiply the average base by the height.
What is the difference between a trapezoid and a trapezium?
It is mostly a regional difference. In the United States, a quadrilateral with one pair of parallel sides is called a trapezoid. In British English, that same shape is called a trapezium. If a problem says trapezium, check which convention the book uses: the British trapezium and the American trapezoid are the same shape, with the same area formula.
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