Trapezoid
A trapezoid is a quadrilateral with exactly one pair of parallel sides, called the bases.
Formula
\text{Area} = (1/2)(b_1 + b_2) \times \text{height}
Definition
A trapezoid is a four-sided shape with exactly one pair of parallel sides, called the bases (one longer, one shorter), with the other two sides called legs. Area $=(1/2)(b_1+b_2)\times$height, and the midsegment (connecting the midpoints of the legs) has length $(b_1+b_2)/2$ and is parallel to both bases; co-interior angles between each leg and the parallel bases are supplementary. Any trapezoid is affinely equivalent to a right trapezoid, and its area formula can be derived by decomposing it into a rectangle and two triangles.
Example
The shape of a bucket, a barn cross-section, or a trapeze platform is a trapezoid; if one base is $10$ cm, the other $6$ cm, and the height $4$ cm, the area is $(10+6)/2\times4=32$ square cm. With bases $12$ and $8$, height $5$: area $=(1/2)(20)(5)=50$, midsegment $=10$, and if one base angle is $70^\circ$, the co-interior angle on the same leg is $110^\circ$. For vertices $A(0,0)$, $B(8,0)$, $C(6,4)$, $D(2,4)$: bases $AB=8$, $DC=4$, height $4$, giving area $(1/2)(8+4)(4)=24$, confirmed by the shoelace formula.
Key Insight
The area formula averages the two bases and multiplies by the height, since the trapezoid sits "between" two rectangles, one with each base. The midsegment being the average of the two bases connects to the midsegment theorem for triangles (midsegment = half the base), since a triangle is a degenerate trapezoid where one base shrinks to zero; the same formula also underlies the trapezoidal rule for numerical integration, where the "bases" become function values and the "height" the interval width, linking elementary geometry to calculus.