Area of a Trapezoid
Geometry & MeasurementThe area of a trapezoid is one-half the sum of its two parallel sides multiplied by the height between them.
Formula
A = \frac{1}{2}(b_1 + b_2) \times h
Definition
A trapezoid has two parallel sides called bases. To find its area, add the two bases together, multiply by the height, then divide by $2$.
Example
A trapezoid with bases $6$ cm and $10$ cm and height $4$ cm: $A = (1/2)(6 + 10)(4) = (1/2)(16)(4) = 32$ cm². You average the two bases and multiply by the height.
Key Insight
The formula averages the two parallel sides. Think of it as finding the area of a rectangle whose width is the average of the two bases, times the height.
Definition
A trapezoid (trapezium in British English) has exactly one pair of parallel sides, called bases $b_1$ and $b_2$, separated by perpendicular height $h$. Its area is $A = (1/2)(b_1 + b_2)h$. This can be derived by splitting the trapezoid into two triangles sharing the height $h$.
Example
A highway cross-section is trapezoidal: top width $20$ m, bottom width $30$ m, depth $3$ m. Area $= (1/2)(20 + 30)(3) = 75$ m$^2$. Multiply by road length to get volume of material.
Key Insight
The trapezoid formula is a generalization: if $b_1 = b_2$, it becomes the parallelogram formula ($b \times h$); if $b_2 = 0$, it becomes the triangle formula $(1/2) \cdot b \cdot h$. One formula covers rectangle, parallelogram, triangle, and trapezoid.
Definition
The trapezoid area formula $A = (1/2)(b_1 + b_2)h$ is the simplest case of numerical integration by the trapezoidal rule, which approximates the definite integral of $f$ on $[a,b]$ as $(b-a)(f(a)+f(b))/2$. The midsegment (median) of a trapezoid has length $(b_1+b_2)/2$, so $A = \text{midsegment} \times h$.
Example
The trapezoidal rule applied to $f(x) = x^2$ on $[0,2]$ gives $(2)(0+4)/2 = 4$, vs. the true integral of $8/3 = 2.67$. The error is $O(h^2)$, where $h$ is the step size, making it more accurate than the rectangle rule.
Key Insight
The midsegment connection explains why the trapezoidal rule works: the area under a curve is approximated by a trapezoid whose parallel sides are the function values at the endpoints, and the midsegment is the arithmetic mean of the two function values.