Quadrilaterals

Parallelogram

General parallelogram (Rhomboid)

A parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. A parallelogram with adjacent sides unequal is called rhomboid.

Rhombus

A rhombus is a parallelogram with four sides of equal length. The diagonals of a rhombus are perpendicular with each other.

Rectangle

A rectangle is a parallelogram with four angles of equal size (right angles).

Square

A square is a parallelogram with four sides of equal length and angles of equal size (right angles).

Quadrilateral

General quadrilateral

A quadrilateral is a polygon with four edges (sides) and four vertices (corners).

Area with known four sides and one angle say "α":

The area can be divided into two triangles. Area A₁ is a triangle with two sides and an included angle known, while A₂ is a triangle with three sides known

Area with known four sides and sum of opposite angles.

Area with known diagonals d₁ and d₂ and their intersection angle β.

Cyclic quadrilateral

A cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle.

The sum of opposite angles of a cyclic quadrilateral sum up to 180°.

A + C = 180° and B + D = 180°

The radius of the circumscribing circle is:

Ptolemy's theorem expresses the product of the lengths of the two diagonals d₁ and d₂ of a cyclic quadrilateral as equal to the sum of the products of opposite sides.

For a cyclic quadrilateral with successive sides a, b, c, d, semiperimeter s, and angle A between sides a and d:

Trapzoid or Trapzium

Trapezoid (or trapezium) is a convex quadrilateral with at least one pair of parallel sides.

If the consecutive sides are a, c, b, d and the diagonals are p, q:

The distance e between the midpoints of the diagonals is:

Length of diagonals, where a is the short base and b is the long base:

Dividing line of trapezoid

If a trapezoid is divided by a line parallel to the parallel sides and such that the ratio of the areas is

the length x of the dividing line ef is:

If A₁ = A₂

This the the quadratic mean or root mean square (RMS) of the the bases of the trapezoid.