A quadrilateral is a geometric figure having four sides and four angles which always total 360°. We will discuss all types of quadrilaterals except the concave quadrilateral. (See diagram). This type of quadrilateral has one angle greater than 180°. (Angles greater than 180° are called concave angles). These quadrilaterals are not discussed much in a typical geometry course and are not among the quadrilaterals with which you are familiar. Generally, all a quadrilateral needs to be classified as such is four sides. However, there are six specific quadrilaterals that are worth discussing in detail.
Click here for a trapezoid calculator. The British use the term trapezoid to refer to a quadrilateral with no parallel sides and a trapezium is a quadrilateral with two parallel sides. The American usage is the exact opposite of the British usage: trapezoid  two parallel sides trapezium  no parallel sides. The only requirement for a trapezoid (American definition) is that two sides are parallel. Side a and side c are the parallel sides and are called bases. The nonparallel sides (side b and side d) are called legs. Lines AC and BD are the diagonals. The median is perpendicular to the height and bisects lines AB and CD. ∠ A plus ∠ B = 180° ∠ C plus ∠ D = 180° Trapezoid Area = ½ • (sum of the parallel sides) • height height² = [(a +b c +d) • (a +b +c +d) • (a b c +d) • (a +b c d)] ÷ (4 • (a c)²)
Two special cases of trapezoid are worth mentioning.
The legs and diagonals of an isosceles trapezoid are equal.
Both pairs of base angles are equal The right trapezoid has two right angles.
Click here for a kite calculator. ∠ A and ∠ D are vertex angles ∠ B = ∠ C and are the nonvertex angles Lines AD and BC are diagonals and always meet at right angles. Diagonal AD is the axis of symmetry and bisects diagonal BC, bisects ∠ A and ∠ D, and bisects the kite into two congruent, triangles. (△ ABD and △ ACD) Diagonal BC bisects the kite into two isosceles triangles. (△ ABC and △ BCD) Side AB = side AC, side BD = side CD and Line OB = Line OC
Click here for a parallelogram calculator.
• Both pairs of opposite sides are equal. (Side AD = BC Side AB = DC)
Click here for a rhombus calculator.
• All four sides are equal (Side AB = BD = DC = CA)
Click here for a rectangle and square calculator. • Opposite sides are parallel and equal • All 4 angles are right angles • Diagonals bisect each other and are equal • Rectangle Area = length × width • Perimeter = (2 × length) + (2 × width)
Click here for a rectangle and square calculator. • All 4 sides are equal • All 4 angles are right angles • Diagonals bisect each other at right angles and are equal • Perimeter = 4 × side length • Area = (side length)^{2}
A circle can be circumscribed around a cyclic quadrilateral and its opposite angles add up to 180°
A circle can be inscribed inside a tangential quadrilateral.
