Composite Figures
composite_figures.gif

What is a composite Figure?


A composite figure is a 2-D figure that is attached (combined) to other 2-D figures. These combined figures will yield unique measurements.


external image 9-3-composite-1.jpg
For example: This would be considered a composite figure, because it contains more than one shape. In this figure, a rectangle and a square are both present.

Perimeter - The distance around the outside of an object.
(the red area represents where you would find each measurement)
Example: If you were fencing in a backyard, you would need to find the perimeter of the space you are fencing.

square.jpg



Explanation: For perimeter, you add up all of the side lengths. If each side length was 5cm, it would be
P=5+5+5+5
P= 20cm
Therefore, the perimeter of the square would be 20 cm.

Area - The number of square units that make up a shape.
Example: If you were covering a lawn with new sod you would want to know the area of the lawn being covered.

square_representing_area.jpg

Explanation: To find the area of a square, use the formula A=lw. If each side is 10m, it would be
A=(10)(10)
A=100m²
Therefore, the area of the square would be 100m².

Surface Area - The number of square units covering a 2 or 3-D shape.
Example: If you were painting a bedroom and you needed to know how many cans of paint to buy, the surface area would have to be found.

cube_representing_surface_area.jpg

Explanation: To find the surface area of a cube, use the formula SA=2(wh+lw+lh). So if each side is 1 mm long, it would be
SA= 2(wh+lw+lh)
SA= 2(1x1+1x1+1x1)
SA= 2(1+1+1)
SA= 2(3)
SA= 6 mm
Therefore, the surface area of the cube is 6 mm².

Volume - The number of cubic units that fill a 3-D shape. Volume can also be referred to as capacity.
Example: If you needed to know how much water your bath tub could hold.
cube_representing_surface_area.jpg
Explanation: To find the volume of a cube, use the formula V=lwh. If each side is 2 km long, it would be
V=(2)(2)(2)
V= 8 km*cubed*
Therefore, the volume of the cube is 8 km*cubed*.

2-D Shapes vs. 3-D Shapes - A 2-D (2 dimensional shape) does not have any depth, and is only one face of a shape. A 3-D (3 dimensional) shape has a depth, and shows all sides of a shape.

external image shape-triangle.gifexternal image tri_prism.jpg



(2-D shape on left) (3-D shape on right)



Net Diagrams - A 2-D diagram which shows the total surface area of a 3-D shape. These are necessary when finding the surface area, area, or volume of a 3-D shape.

external image 400px-Necker_cube.svg.pngexternal image cube.gif
(3-D shape/cube on left) (Net-Diagram of Cube on right)


Formulas of Perimeter and Area/Surface Area for 2-D shapes
Perimeter Formulas:
Rectangle-P=l+l+w+w or P=2(l+w)
Parallelogram-P=b+b+c+c or P=2(b+c)
Triangle-P=a+b+c
Trapezoid-P=a+b+c+d
Circle-C=πd or 2πr

Area/Surface Area Formulas:
Rectangle-A=lw
Parallelogram-A=bh
Triangle-A=bh/2 or A=1/2bh
Trapezoid-A=(a+b)h/2 or A=1/2(a+b)h
Circle-A=πr²


Formulas of Area/Surface Area and Volume for 3-D shapes
Area/Surface Area Formulas:
Cylinder-A=2πr²+2πrh
Sphere-A=4πr²
Cone-A=πrs+πr²
Square-based pyramid-A=2bs+b²
Rectangular prism-A=2(wh+lw+lh)
Triangular prism-A=(ls+lb+lh)+bh

Volume Formulas:
Cylinder-V=πr²h
Sphere-V=4/3πr3 or V=4πr3/3
Cone-V=1/3πr²h or V=πr²h/3
Square-based pyramid-V=1/3b²h or b²h/3
Retangular prism-V=lwh
Triangular prism-V=1/2bhl or V=bhl/2


Area:


To find the area of a composite figure, you must recognize all the different shapes in the figure and find the area of those shapes and then add them together. Ex1:

external image Image15582.gif

This shape is a composite figure and the area needs to be found. So, we divide it into 2 shapes, a rectangle and a triangle. When we divide it, it looks like this:

external image Image15583.gif

Rectangle = A1
Triangle = A2

A1 = (L)(W)
= (14)(12)
= 168cm squared

A2 = 1/2(b)(h)
=1/2(8)(12)
=48cm squared

Total Area = A1 + A2
=168 + 48
=216cm squared

That's how you find the area of a composite figure. Sometimes there will be more than one shape but there is no difference in what you do to find the area.

Practical Example of Area:

Hank wants to lay hardwood in his family room. When he goes to take the measurements of the room, he realizes that it is a composite figure. In order to buy the right amount of hardwood, he must figure out the area of the room. This is the shape and measurements of Hank's family room:

Hank's_room.jpg

Find the area of this room.

1. Recognize all of the different shapes within Hank's room.

This is a composite figure containing two rectangles. Hank must find the area of both, separately, and then add them together to get the total area of his family room.


Hank's_room_numbered.jpg

2. Find the area of each figure and add them together to discover the total area.

Rectangle 1 = A1
Rectangle 2 = A2

A1 =
(L)(W)
A1 = (6)(3)
A1 = 18m squared
A1 = 18m²
Therefore, the area of rectangle 1 is 18m².

A2 = (L)(W)
A2 = (15)(3)
A2 = 45m squared
A2 = 45m²
Therefore, the area of rectangle 2 is 45m².

Total Area = A1 + A2
=18 + 45
= 63m squared
= 63m²

The total area of Hank's family room is 63m², which means that he must buy 63m² of hardwood in order to cover the whole floor.
Perimeter:

The perimeter is found by adding all the measurements on the outside of the figure. Do not measure the inside line of the shape if there is one. This does not count as a measurement for the perimeter.


(Each square = 1cm)

P = S1 + S2 + S3 + S4 + S5 + S6 + S7 + S8
=2 + 1 + 3 + 4 + 8 + 3 + 3 + 2
=26cm

Practical Example of Perimeter:
Mary is putting a pool into her back yard. The regulations say that she must put a fence around her back yard if she is going to have a pool. Mary is going to buy fencing for her back yard, but does not know how much to buy, because it is a composite figure. This is what Mary's back yard looks like:

Mary's_Back_yard.jpg

Find the perimeter of Mary's back yard.

1. Add up each side measurement in order to find the total perimeter.

S1 = 3m
S2 = 9m
S3 = 16m
S4 = 5m
S5 = 3m
S6 = 12m
S7 = 10m
S8 = 8m

P = S1 + S2 + S3 + S4 + S5 + S6 + S7 + S8
=
3 + 9 + 16 + 5 + 3 + 12 + 10 + 8
=66m

Therefore, the total perimeter of Mary's back yard is 66m, which means that she needs to buy 66m of fencing to enclose her whole back yard.

Practical Examples of Volume:
Brian wants to find out the volume of his swimming pool so that he can figure out how much time it would take for the water to fill up the pool. The water pump pumps 0.3m² of water per minute. This is how his swimming pool looks like:
wiki.JPG
The Pool is 5m Wide. Find the Volume.
1. First, divide the figure into three to make it a composite figure. Then find and fill in the missing measurements for the composite figures.

wiki2.JPG
2. After this step is done, just figure the volume of the 3-D shapes and add them up.
*Remember*
Formulas of Volume for 3-D figures.
Rectangular Prism-lwh
Triangular Prism-bhl/2
V=lwh+bhl/2+lwh
V=(12)(5)(1)+(5)(2)(5)/2+(3)(5)(2) In this case, the twos cancel out in the formulas of the Triangular Prism.
V=60+25+30
V=115m3
Therefore the volume of the pool is 115m3.
3. Now to find out how long it would take for the pool to fill, divided the volume of the pool by 0.3. Then divide the outcome by 60 to get the time in hours.
115/0.3=383.3333333minutes.
383.3333333/60=6.3888888889 hours.
Therefore it would take around 6 hours and 39 minutes for the pool to fill up.

So, why is the topic of Composite Figures important anyways?
So, why is this topic really important? Will I even need it after school? First of all, no matter what career path you decide to take you will most likely use surface area, area, volume, etc. at least once in your life. Whether you are trying to find the area of your backyard to put down patio stones or are trying to find the volume of your community pool, all these formulas will come in handy! These calculations are used daily, and will be useful in your future!





Try out some more perimeter and area problems at:


http://mathscore.com/math/free/lessons/Texas/8th_grade/Perimeter_and_Area_of_Composite_Figures_sample_problems.html
math score

Watch how to solve these problems at:

http://www.youtube.com/watch?v=0lcfV-1Qh8w

If you want a video example done, click below to watch the video:


If you want a Powerpoint example done, click the link to open the file.





Citations:

Cube. (n.d.). [On-line Photograph]. Retrieved on May 16, 2009, from rationalwiki.com

Composite figures Powerpoint (June 4, 2009)

Composite figures Title (June 4, 2009)

Definitions- Mr.T.Page

Formulas of Perimeter, Area/Surface Area, and Volume of 2-D and 3-D shapes - Mr. T. Page (Tuesday, May 26, 2009)

Net Diagram of Cube. (n.d.). [On-line Photograph]. Retrieved on May 16, 2009, from gwydir.demon.co.uk/ jo/solid/cube.htm

Practical Examples of Volume - Mr. Page, Mathpower 9 Textbook (Sunday, May 24, 2009)

Red Triangle. (n.d.). [On-line Photograph]. Retrieved May 16, 2009, from
http://www.indstate.edu/cirt/ittrain/resources/tutorials/instructional/hotpotatoes/shape-triangle.gif

Shape Used in the Practical Example for Area. (n.d.). [On-line Photograph]. Retrieved on May 24, 2009, from
http://mathscore.com/math/free/sampleProblems/img_614.gif

Triangular Prism. (n.d.). [On-line Photograph]. Retrieved May 16, 2009, from www.blairstownelem.net

Video Lesson on Composite Figures-Youtube. (March 24, 2009). [On-line Video]. Retrieved on May 25, 2009, from
http://www.youtube.com/watch?v=0lcfV-1Qh8w

Composite Figures Title. (June 3, 2009)

Composite Figures Powerpoint Lesson. (June 4, 2009)