Trapezoid
GeometryA 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. The two parallel sides are called bases (one is longer, one is shorter), and the other two sides are called legs.
Example
The shape of a bucket, a barn cross-section, or a trapeze platform is a trapezoid. If one base is $10$ cm and the other is $6$ cm, and the height is $4$ cm, the area $= (10 + 6)/2 \times 4 = 32$ square cm.
Key Insight
The area formula for a trapezoid averages the two bases and multiplies by the height. This makes sense: the trapezoid is "between" two rectangles, one with the smaller base and one with the larger base.
Definition
A trapezoid (US usage) has exactly one pair of parallel sides (the bases, $b_1$ and $b_2$) and two non-parallel sides (the legs). Area $= (1/2)(b_1 + b_2)h$. 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.
Example
Trapezoid with bases $12$ and $8$, height $5$: Area $= (1/2)(20)(5) = 50$. Midsegment $= (12+8)/2 = 10$. If one base angle is $70^\circ$, the co-interior angle on the same leg is $110^\circ$.
Key Insight
The midsegment of a trapezoid is the average of the two bases - a beautiful result. This connects to the midsegment theorem for triangles (midsegment = half the base), since a triangle is a degenerate trapezoid where one base shrinks to zero.
Definition
A trapezoid is a quadrilateral with exactly one pair of parallel sides (using the exclusive US definition). Area $= (1/2)(b_1+b_2)h$ can be derived by decomposing into a rectangle and two triangles, or by shear: any trapezoid is affinely equivalent to a right trapezoid, preserving area ratios.
Example
A trapezoid with vertices $A(0,0)$, $B(8,0)$, $C(6,4)$, $D(2,4)$: bases $AB=8$ ($y=0$) and $DC=4$ ($y=4$), height$=4$. Area$=(1/2)(8+4)(4)=24$. By shoelace: $(1/2)|0\cdot0-8\cdot0+8\cdot4-6\cdot0+6\cdot0-2\cdot4+2\cdot0-0\cdot4|=(1/2)|0+32+0-8+0|=24$.
Key Insight
The trapezoid area formula is a special case of the trapezoidal rule for numerical integration: the area under a curve on $[a,b]$ is approximated by $(b-a)(f(a)+f(b))/2$. The "bases" become function values and the "height" becomes the interval width, connecting elementary geometry to calculus.