Geometry » Quadrilaterals

Contents

- Scalene or irregular quadrilateral
- Square
- Rectangle
- Parallelogram
- Kite
- Rhombus
- Trapezium (US: Trapezoid)
How does the set of quadrilaterals relate to each other?

Scalene or irregular quadrilateral

A quadrilateral without any special properties. There is no symmetry and there are no lines parallel or perpendicular to each other.
scalene quadrilateral

Square (regular quadrilateral)

A square is a quadrilateral with four equal sides and four equal angels.
The diagonals intersect perpendicularly and bisect each other.
The diagonals are equally long.
There are two pairs of equal sides.
square

Area = length × width
= side × side
= side2

Rectangle

A quadrilateral with four right angles (90°).
In a rectangle the diagonals bisect each other.
The diagonals are equally long.
Rectangle
Area = length × width

Parallelogram

A quadrilateral with rotation symmetry.
A quadrilateral with two pairs of parallel lines.
The opposite sides are equal in length (and parallel).
The opposite angles are equal in size.
The diagonals cut each other into two equal parts.
Parallelogram
Area = base × height

Example
Parallelogram with sides of 10 and 8 and a height of 6 drawn perpendicular on the side of 10
Base and height always are perpendicular to each other.
Area ABCD = 10 × 6 = 60

Kite

A kite is a quadrilateral where a diagonal is an axis of symmetry.
Two by two the sides are equal in length.
The diagonals intersect perpendicularly.
Two angles have the same size.
kite
Area = length diagonal PR × length diagonal QS : 2

Example
Kite ABCD with S as intersection of the diagonals. BC=15, CS=8 en CD=10
We need the length of both diagonals.
AC = 2 × 8 = 16
The length of BD is not given, so we need to calculate it.
We can do this with Pythagoras' theorem.

sidesquare
CS = 8  64 
DS = ?36+  
CD = 10100

 DS = square root(36) = 6

    
sidesquare
CS = 8  64 
BS = ?161+  
BC = 15225

 BS = square root(161)

 BD = DS + BS = 6 + square root(161)
 Area ABCD = AC × BD : 2 = 16 × (6 + wortel 161) : 2 ≈ 149.5

Rhombus

A rhombus is a quadrilateral with four equal sides.
A rhombus is a quadrilateral where both diagonals are axes of symmetry.
The opposite angles are equal in size.
The diagonals intersect perpendicularly and bisect.
The diagonals divide the angles into two equal parts (angle bisector).
rhombus
Area = see kite

Trapezium (US: trapezoid)

A quadrilateral with one pair of parallel lines.
No special properties.
trapezium/trapezoid
Area = (length small base + length large base) × height : 2

Example
Trapezium ABCD in which CS is the height, perpendicular to AB. AS=10, BS=16, BC=20, CD=6 en CS=12
Base and height always are perpendicular to each other.
AB = AS + BS = 10 + 16 = 26
Area ABCD = (AB + CD) × CS : 2 = (26 + 6) × 12 : 2 = 192

Isosceles trapezium

When the base angles in a trapezium are equal, the trapezium is isosceles.
Has one axis of symmetry.
Has two pairs of equal angles.
Two sides (drawn blue underneath) are equal in length.
isosceles trapezium
Area = See trapezium

How does the set of quadrilaterals relate to each other?

See the figure below:
Figure that shows the information/set underneath in arrows

A square is a special rectangle.
A square is a special rhombus.
A square is a special parallelogram.
A square is a special kite.
A square is a special trapezium.
A rectangle is a special parallelogram.
A rectangle is a special trapezium.
A rhombus is a special parallelogram.
A rhombus is a special kite.
A rhombus is a special trapezium.
A parallelogram is a special trapezium.


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