# Formulas

## 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: $$S$$
1. 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.
2. 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.
3. The midline of a trapezoid is parallel to the bases and equal to the arithmetic mean of the lengths of the bases.
$$m = {\large\frac{{a + b}}{2}\normalsize},\;$$ $$m\parallel a,\;$$ $$m\parallel b$$
4. Diagonals of a trapezoid (if $$a \gt b$$)
$$p = \sqrt {\large\frac{{{a^2}b – a{b^2} – b{c^2} + a{d^2}}}{{a – b}}\normalsize},\;$$ $$q = \sqrt {\large\frac{{{a^2}b – a{b^2} – b{d^2} + a{c^2}}}{{a – b}}\normalsize}$$
5. Perimeter of a trapezoid
$$P = a + b + c + d$$
6. Area of a trapezoid
$$S = {\large\frac{{a + b}}{2}\normalsize} h = mh$$
$$S =$$ $${\large\frac{{a + b}}{2}\normalsize}$$ $$\sqrt {{c^2} – {{\left[ {\large\frac{{{{\left( {a – b} \right)}^2} + {c^2} – {d^2}}}{{2\left( {a – b} \right)}}\normalsize} \right]}^2}}$$
7. All four vertices of an isosceles trapezoid lie on a circumscribed circle.
$$R =$$ $${\large\frac{{c\sqrt {ab + {c^2}} }}{{\sqrt {\left( {2c – a + b} \right)\left( {2c + a – b} \right)} }}\normalsize}$$
$$p = \sqrt {ab + {c^2}}$$
$$h = \sqrt {{c^2} – {\large\frac{1}{4}\normalsize}{{\left( {a – b} \right)}^2}}$$
$$a + b = c + d$$
$$r = {\large\frac{h}{2}\normalsize},$$
where $$h$$ is the altitude of the trapezoid.