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Quadrilaterals

From the latin root: quadri - four; latus - side Definition: a closed four-sided figure. All of its sides must be straight.
There are many different kinds of quadrilaterals, but all of them have several things in common. They all have four sides and four vertices, and their interior angles all add up to 360 degrees. We know many quadrilaterals by their special shapes and properties, like squares. Remember, if you see the word quadrilateral, it does not necessarily mean a figure with special properties like a square or rectangle! In word problems, be careful not to think that a quadrilateral has parallel sides or equal sides unless that is stated.

Regular Quadrilaterals:

Rectangle

A rectangle is a quadrilateral with all of its interior angles equal to 90°. The opposite sides of a rectangle run parallel to each other. A rectangle can also go under the parallelogram catergory. A square is a type of rectangle.


Area of a Rectangle
Formula:
A=lw


Example:
external image 364px-Prostokat-rectangle.svg.png

a=3cm
b=5cm

A=lw

=5x3
=15cm²

Therefore, the area of the rectangle is 15cm².

Perimeter of a Rectangle
Formula;
P=2(l+w)


Example:

external image 364px-Prostokat-rectangle.svg.png

a=3cm
b=5cm

P=2(l+w)
=2(3+5)
=2(8)
=16cm

Therefore, the perimeter of the rectangle is 16cm.

Squares

A square is a quadrilateral with all four of it's sides and angles equal. All of it's interior angles are 90°. Another quality of a square is that if you were to draw lines connecting opposet vertexes (corners) the diagonals would intercept at a 90° angle (or perpendicular). A square can be considered as a rectangle, parallelogram, or rhombus.

Area of a Square:

Formula:
A=s²

Example:

external image Square_1000.gif

a=10cm

A=s²
=10²
=100cm²

Therefore, the area is 100cm²
.


Perimeter of a square:

Formula:
P=4s

Example:


external image Square_1000.gif
a=10cm

P=4s
=4x10
=40cm

Therefore, the perimeter is 40cm.


Parallelograms
Parallelogram
Parallelogram
Opposite sides of a parallelogram are parallel and equal in length. The opposite angles are congruent. Angles "a" are the same, and angles "b" are the same. Squares, Rectangles and Rhombuses are all Parallelograms.

Any side of a parallelogram can be the base. Once you have the base, the height of the parallelogram is the length of any line segment perpendicular from the base parallel to the opposite side. Also, adjacent angels of a parallelogram always have a sum of 180 degrees. Each diagonal bisects one another.

The area of a parallelogram is A = b x h, "b" representing the length and "h" representing the height.
Example:
A=?
b=10cm
h=5cm
A = b x h
A = 10 x 5
A = 50cm²
The area is 50 cm².

The perimeter of a parallelogram is the distance around the outside of it. The formula is side + side + side + side, since a parallelogram has four sides. With opposite sides being congruent, you could use the formula 2s+2s as long as you used two different sidelengths.
Example:

external image rectangle_8x3.gif
P = S+S+S+S
P = 8 cm + 8cm + 3 cm + 3 cm = 22 cm
Therefore, the perimeter is 22cm.



Rhombus

A rhombus is a different type of parallelogram. Instead of having opposite sides parallel and equal in length, all the sides of a rhombus have equal length. A rhombus has some of the same characteristics of a square but not all angles need to be 90°. All sides of a rhombus and be named the base because they are all equal.

Area of a rhombus:
A=ba

A= base × altitude (altitude is also known as height)


external image rhombus.gifexternal image tp2ch3_images1.jpg


Trapezoids

A trapezoid is a quadrilateral with one pair of parallel sides. The pair of parallel sides are not the same length, but the other two sides are. A trapezoid also has two obtuse angles, and two acute angles.

Area of a Trapezoid:

Formula:
A= h[(b1+b2)/2]


Example:
external image isoscelestrapezoid.gif
a= 10cm
b=9cm
c=5cm
height: 4cm

A= h[(b1+b2)/2]

= 4[(10+9)/2]
= 4(19/2)
= 4x19.5
= 38cm²

Therefore, the area of this trapeziod is 38cm²
.

Perimeter of a Trapezoid:

Formula:
P= s+s+s+s

Example:

external image isoscelestrapezoid.gif

a= 10cm
b=9cm
c=5cm

P=s+s+s+s
=10+9+5+5
=29cm

Therefore, the perimeter of the trapezoid is 29cm
.


Quadrilaterals - How Do They Relate To The Real World?

Many of our everyday items are quadrilaterals. Example. paper - rectangle, kite - rhombus, piece of cheese - square. :P
Quadrilaterals are all around us so we need to know about them!

Why Do We Need To Know The Perimeter of Quadrilaterals?

There are some problems that we may come across, in which case we need to know how to solve them by learning about the perimeter of quadrilaterals.

Example:

Harley lives in a house with a square yard around it. Each side of her yard is 20m. She wants to build a fence around it, and her house (which is inside it). How much fencing will she need?

-->To figure this out, we need to know the perimeter of her yard because that will tell us how much fencing she needs. To find the perimeter we need to know how to figure it out! Because we know the perimeter, this is how we find the answer:

P=4s
=4x20m
=80m

Therefore, Harley needs 80m of fencing because the perimeter of her backyard is 80m.


Why Do We Need To Know The Area Of Quadrilaterals?

In real life, there are lots of problems, and jobs, that deal with area. So, to figure them out, we need to know how!

Example:

Harley's house has a rectangular bathroom. The dimensions of her bathroom is 3m by 4m. Harley wants to cover the whole bathroom floor with tiles. How much tiles does she need?

-->To figure this out, we need to know the area of Harley's bathroom, because that will tell us how much tiling she will need to cover it. So, we use our area formula for a rectangle to get the answer:

A=lxw
=3x4
=12m²

Therefore, Harley needs 12m² of tiling to cover her whole bathroom floor.

Irregular Quadrilaterals:

Irregular quadrilaterals still have four sides and four angles. Here are the differences between a regular and irregular quadrilateral:

REGULAR QUADRILATERALS
IRREGULAR QUADRILATERALS
- all four of its angles are convex (less than 180°)
-can have concave angles (more than 180°) -have at least pme set of parallel sides
- do not have any parallel sides-can have some or all of their side lengths the same
- all of their side lengths can be different


Here is an example of an irregular quadrilateral:

external image 114027.png
It is an irregular concave square. This is BECAUSE...
- it has a concave angle
- it has no parallel sides


    • EVEN THOUGH SOME OF ITS SIDE LENGTHS ARE EQUAL, IT IS STILL IRREGULAR BECAUSE OF ITS OTHER ATTRIBUTES

Some Other Pages That Might Help Your Understanding:

Angles
Composite Figures
GeometryOptimal ValuesParallel and Perpendicular LinesPolygons
Citations**
Page, John (2007) Rhombus. Math Open Reference 2007. Retrieved December 13, 2008 from: http://www.mathopenref.com/rhombus.html
Pierce, R. (n.d.). Quadrilaterals. Retrieved November 17, 2008, from http://www.mathsisfun.com/quadrilaterals.html
Sanchez, P. (2007, July 18). Parallelograms. Retrieved December 16, 2008, from http://planetmath.org/encyclopedia/Parallelogram.html

Page, John. (2007) Trapezoid. Math Open Reference 2007. Retrieved December 17, 2008 from: **http://www.mathopenref.com/trapezoid.html**
Page, John. (2007) Square. Math Open Reference 2007. Retrieved December 17, 2008 from: http://www.mathopenref.com/square.html
Weisstein, Eric W.(1999-2008) Square. Wolfram MathWorld. Retrieved December 17, 2008 from: http://mathworld.wolfram.com/Square.html
Page. John. (2007). Quadrilaterals. Math Open Reference 2007. Retrieved December 18, 2008 from: http://www.mathopenref.com/quadrilateral.html
Page, John. (2007). Irregular Polygons. Math Open Reference 2007. Retrieved December 18, 2008 from: http://www.mathopenref.com/polygonirregular.html
Page, John. (2007). Perimeter of a Rectangle. Math Open Reference 2007. Retrieved December 18, 2008 from: http://www.mathopenref.com/rectangleperimeter.html
Page, John. (2007). Rectangle. Math Open Reference 2007. Retrieved December 18, 2008 from: http://www.mathopenref.com/rectangle.html



THIS PAGE WAS FINISHED BY: NICK, HARLEY, PAIGE & KAYLA :)