QUADRILATERALS

Q u a d r i l a t e r a l s Scroll to the bottom for a listing of
properties for all types of quadrilaterals

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.

Trapezoids
Click here for a trapezoid calculator.

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°

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.

Kites
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

Parallelograms
Click here for a parallelogram, rhombus, rectangle and square calculator.

• Both pairs of opposite sides are equal. (Side AD = BC Side AB = DC)
• Both pairs of opposite angles are equal (∠ A = ∠ C ∠ D = ∠ B)
• Diagonals bisect each other.
• Each diagonal forms two congruent triangles.
Diagonal AC forms △ ADC and △ ABC
Diagonal DB forms △ DCB and △ DAB
• Parallelogram area = Line AB × height or area = Line DC × height.

Rhombuses

• All four sides are equal (Side AB = BD = DC = CA)

• Diagonals meet at right angles

• Diagonals bisect each other

• Both pairs of opposite angles are equal
∠ A = ∠ D ∠ C = ∠ B

• Rhombus Area = (AD × CB) ÷ 2

Rectangles

• 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)

Squares

• 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

Quadrilateral Properties


D I A G O N A L S	Isosceles Trapezoid	Kite	Parallel ogram	Rhombus	Rectangle	Square
Diagonals Bisect Each Other			Y E S	Y E S	Y E S	Y E S
Diagonals Are Equal	Y E S				Y E S	Y E S
Diagonals Meet At Right Angles		Y E S		Y E S		Y E S
P A R A L L E L S I D E S	Isosceles Trapezoid	Kite	Parallel ogram	Rhombus	Rectangle	Square
Only 2 Sides Are Parallel	Y E S
No Sides Are Parallel		Y E S
Both Pairs of Opposite Sides Are Parallel			Y E S	Y E S	Y E S	Y E S
E Q U A L S I D E S	Isosceles Trapezoid	Kite	Parallel ogram	Rhombus	Rectangle	Square
Only 2 Sides Are Equal	Y E S
Both Pair of Adjacent Sides Are Equal But No Opposite Sides Are Equal		Y E S
Both Pair of Opposite Sides Are Equal			Y E S	Y E S	Y E S	Y E S
All Four Sides Are Equal						Y E S
A N G L E S	Isosceles Trapezoid	Kite	Parallel ogram	Rhombus	Rectangle	Square
Both Pairs of Base Angles Are Equal	Y E S
Non-Vertex Angles Are Equal		Y E S
Only Opposite Angle Pairs Are Equal			Y E S	Y E S
All Four Angles Are Right Angles					Y E S	Y E S

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