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

  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.

    isosceles trapezoid with a circumscribed circle

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

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

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

  11. 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:
    \(a + b = c + d\)

    trapezoid with an inscribed circle

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

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