Key takeaway: The area of a trapezoid equals half the sum of its two parallel bases times its height, which is the same as the height multiplied by the average of the bases.
Grade 66.G.A.1Geometry and Measurement
Definition of Area of a Trapezoid
The area of a trapezoid measures the flat space inside a four-sided figure that has exactly one pair of parallel sides, called the bases. The other two sides, the legs, may lean.
Here is a picture proof. Make a second copy of the trapezoid, turn it half a revolution, and fit it against a leg. Together the pieces form a parallelogram whose base is the two bases laid end to end and whose height has not changed. That parallelogram holds twice as much surface as one trapezoid, so a single trapezoid holds half. A second way to see it: the average of the two bases is the width of a rectangle with the same surface.
Formula and notation
Using words for the three measurements that matter:
\[A = \frac{1}{2} \times (\text{top base} + \text{bottom base}) \times \text{height}\]
The height is the perpendicular distance between the bases, not the length of a leaning leg.
Key facts
- The area of a trapezoid depends only on the two parallel sides and the gap between them.
- Only the two parallel sides are bases, no matter how the figure is turned.
- If both bases are equal, the rule reduces to base times height, as for a parallelogram.
- If one base shrinks to zero, the rule reduces to the triangle formula.
Area of a Trapezoid example
A raised flower bed is a trapezoid whose parallel sides measure 14 feet and 9 feet, with 6 feet of straight-across distance between them. How many square feet of mulch cover does it need?
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Add the bases: \(14 + 9 = 23\).
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Take half: \(\frac{1}{2} \times 23 = 11.5\).
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Multiply by the height: \(11.5 \times 6 = 69\).
The bed needs 69 square feet of cover.
Common mistake
Watch out
Sam computes \(\frac{1}{2} \times 14 \times 9 \times 6 = 378\).
Why it is wrong
He multiplied the two bases instead of adding them. The area of a trapezoid needs the sum of the bases, \(14 + 9 = 23\), then half of that times the height: \(\frac{1}{2} \times 23 \times 6 = 69\). A sanity check: the bed fits inside a 14 by 6 rectangle of area 84, so the answer must be below 84.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Geometry and Measurement