Isosceles Trapezoid
An isosceles trapezoid is a trapezoid with equal legs, equal base angles, and congruent diagonals.
Formula
\text{Area} = (1/2)(b_1 + b_2) \times \text{height}
Definition
An isosceles trapezoid is a trapezoid whose two non-parallel legs are equal in length, making it symmetrical, a mirror image across its line of symmetry (the perpendicular bisector of the two bases). Its base angles are congruent, its diagonals are congruent, and it is always a cyclic quadrilateral, since its opposite angles are supplementary; for bases $b_1, b_2$ and legs $a$, the diagonal length is $d=\sqrt{a^2+b_1b_2}$.
Example
The shape of a lampshade or an upside-down bucket with equal-length slanted sides is an isosceles trapezoid; folding it in half down the middle matches both halves exactly. With bases $10$ and $6$ and legs $5$: height $=\sqrt{5^2-2^2}=\sqrt{21}$, area $=(1/2)(16)(\sqrt{21})=8\sqrt{21}\approx36.7$. For $b_1=10$, $b_2=4$, $a=5$: diagonal $d=\sqrt{25+40}=\sqrt{65}$.
Key Insight
Like an isosceles triangle, an isosceles trapezoid has one line of symmetry, with equal base angles on each base just like the equal base angles of an isosceles triangle. Its cyclic property connects directly to Ptolemy's theorem: for isosceles trapezoid $ABCD$ with equal diagonals $d$ and bases $b_1, b_2$, Ptolemy gives $d^2=b_1b_2+a^2$, confirming the diagonal formula and showing how Ptolemy's theorem unifies both the Pythagorean theorem (rectangle case) and this trapezoid result.