
A trapezoid (or a trapezium) is a quadrilateral in which (at least) one pair of opposite sides is parallel.
Sometimes a trapezoid is defined as a quadrilateral having exactly one pair of parallel sides. The parallel sides are called
the bases, and two other sides are called the legs.

A trapezoid in which the legs are equal is called an isosceles trapezoid.
A trapezoid in which at least one angle is the right
angle (\(90^\circ\)) is called a right trapezoid.

The midline of a trapezoid is parallel to the bases and equal to the arithmetic mean of the lengths of the bases.
\(m = {\large\frac{{a + b}}{2}\normalsize},\;\;m\parallel a,\;\;m\parallel b\)

Diagonals of a trapezoid (if \(a > b\))
\(p = \sqrt {\large\frac{{{a^2}b  a{b^2}  b{c^2} + a{d^2}}}{{a  b}}\normalsize} \), \(q = \sqrt {\large\frac{{{a^2}b  a{b^2}  b{d^2} + a{c^2}}}{{a  b}}\normalsize} \)

Perimeter of a trapezoid
\(P = a + b + c + d\)

Area of a trapezoid
\(S = {\large\frac{{a + b}}{2}\normalsize} h = mh\)
\(S = {\large\frac{{a + b}}{2}\normalsize} \sqrt {{c^2}  {{\left[ {\large\frac{{{{\left( {a  b} \right)}^2} + {c^2}  {d^2}}}{{2\left( {a  b} \right)}}\normalsize} \right]}^2}} \)

All four vertices of an isosceles trapezoid lie on a circumscribed circle.

Radius of the circle circumscribed about an isosceles trapezoid
\(R = \large\frac{{c\sqrt {ab + {c^2}} }}{{\sqrt {\left( {2c  a + b} \right)\left( {2c + a  b} \right)} }}\normalsize\)

Diagonal of an isosceles trapezoid
\(p = \sqrt {ab + {c^2}} \)

Altitude of an isosceles trapezoid
\(h = \sqrt {{c^2}  \large\frac{1}{4}\normalsize{{\left( {a  b} \right)}^2}} \)

If the sum of the bases of a trapezoid is equal to the sum of its legs, all four sides of the trapezoid are tangents to
an inscribed circle:
\(a + b = c + d\)

Radius of the inscribed circle
\(r = \large\frac{h}{2}\normalsize\),
where \(h\) is the altitude of the trapezoid.