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.

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First, it is important to state the difference in definitions between British and American usage.
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 non-parallel 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.
AB = CD and AC = BD

Both pairs of base angles are equal
∠ A = ∠ D and ∠ B = ∠ C

The right trapezoid has two right angles.

Click here for a kite calculator.

∠ A and ∠ D are vertex angles

∠ B = ∠ C and are the non-vertex 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

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• All four sides are equal (Side AB = BD = DC = CA)
• Diagonals meet at right angles
• Diagonals bisect each other
• Diagonals bisect the vertex angles   ∠ A   ∠ B   ∠ C   and   ∠ D
• Both pairs of opposite angles are equal
∠ A = ∠ D     ∠ C = ∠ B
• Adjacent angles are supplementary (they sum to 180°)
∠ A + ∠ B   =   ∠ B + ∠ D   =   ∠ D + ∠ C   =   ∠ C + ∠ A     = 180°
• Rhombus Altitude = side • sine (either angle)   =   (AD • CB) ÷ (2 • side)
• Rhombus Area = (AD × CB) ÷ 2   =   side • altitude   =   side² • sine (either angle)
• Long Diagonal AD = Side Length • Square Root [ 2 + 2 • cos(A) ] • Short Diagonal BC = Side Length • Square Root [ 2 + 2 • cos(B) ] • 4 • Side² = Long Diagonal² + Short Diagonal²

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• 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