# Formulas and Tables

Elementary Geometry# Trapezoid

Bases of a trapezoid: \(a\), \(b\)

Legs of a trapezoid: \(c\), \(d\)

Midline of a trapezoid: \(m\)

Altitude of a trapezoid: \(h\)

Perimeter: \(P\)

Diagonals of a trapezoid: \(p\), \(q\)

Angle between the diagonals: \(\varphi\)

Radius of the circumscribed circle: \(R\)

Radius of the inscribed circle: \(r\)

Area: \(S\)

- 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 \(a \gt b\))

\(p = \sqrt {\large\frac{{{a^2}b – a{b^2} – b{c^2} + a{d^2}}}{{a – b}}\normalsize},\;\) \(q = \sqrt {\large\frac{{{a^2}b – a{b^2} – b{d^2} + a{c^2}}}{{a – b}}\normalsize} \) - 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.