Composite+Figures

Composite Figures
 * __

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.

 __**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.



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


 * __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.



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


 * __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.



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.
 * __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

V=(2)(2)(2) V= 8 km*cubed* Therefore, the volume of the cube is 8 km*cubed*.
 * __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.
 * __Explanation:__** To find the volume of a cube, use the formula V=lwh. If each side is 2 km long, it would be


 * __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.



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

 (3-D shape/cube on left) (Net-Diagram of Cube 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.

__Formulas of Perimeter and Area/Surface Area for 2-D shapes__  __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
 * Perimeter 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
 * Area/Surface Area Formulas:**

__Formulas of Area/Surface Area and Volume for 3-D shapes__  __Cylinder__-A=2 πr² +2 πrh __Sphere__-A=4 πr² __Cone__-A= πrs+πr² __Square-based pyramid__-A=2bs+b² __Rectangular prism__-A=2(wh+lw+lh) __Triangular prism__-A=(ls+lb+lh)+bh
 * Area/Surface Area Formulas:**

__Cylinder__-V= πr²h __Sphere__-V=4/3 πr3 or V=4 πr3 /3 __Cone__-V=1/3 πr²h or V=πr²h/3 __Square-based pyramid__-V=1/3b² h or b²h/3 __Retangular prism__-V=lwh __Triangular prism__-V=1/2bhl or V=bhl/2
 * Volume Formulas:**

__**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:

 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:

 <span style="font-family: 'Comic Sans MS',cursive;">__**Rectangle**__ = A1 <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">**__Triangle__** = A2

<span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">A1 = (L)(W) <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">= (14)(12) <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">= 168cm squared

<span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">A2 = 1/2(b)(h) <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">=1/2(8)(12) <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">=48cm squared

<span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">**__Total Area__** = A1 + A2 <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">=168 + 48 <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">=216cm squared

<span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">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 <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">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:



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.

<span style="font-family: 'Comic Sans MS',cursive;">

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

Therefore, the area of **rectangle 2** is 45m².
 * A2** = (L)(W)
 * A2** = (15)(3)
 * A2** = 45m squared
 * A2** = 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. <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">__**Perimeter**____**:**__

<span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">The perimeter is found by adding all the measurements on the outside of the figure. Do not measure the inside line of the <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">shape if there is one. This does not count as a measurement for the perimeter.

<span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">(Each square = 1cm)

<span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">P = S1 + S2 + S3 + S4 + S5 + S6 + S7 + S8 <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">=2 + 1 + 3 + 4 + 8 + 3 + 3 + 2 <span style="font-family: 'Comic Sans MS',cursive; white-space: pre;"> <span style="font-family: 'Comic Sans MS',cursive;">=26cm

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:
 * Practical Example of Perimeter:**



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

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

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.

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: 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.
 * Practical Examples of Volume:**

2. After this step is done, just figure the volume of the 3-D shapes and add them up. 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.
 * Remember*

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!
 * __So, why is the topic of Composite Figures important anyways?__**

=<span style="color: rgb(255,0,0);">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]<span style="color: rgb(255,0,0);"> <span style="color: rgb(255,0,0);"> <span style="font-size: 120%; color: rgb(255,0,0);">**Watch how to solve these problems at:**

[]

If you want a video example done, click below to watch the video: media type="youtube" key="0lcfV-1Qh8w" height="344" width="425"

If you want a Powerpoint example done, click the link to open the file.
 * __[[file:Composite Figures.ppt]]__**


 * __Citations:__**

Cube. (n.d.). [On-line Photograph]. Retrieved on May 16, 2009, from <span style="color: rgb(0,128,0);">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 []

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 <span style="color: rgb(0,128,0);">www.blairstownelem.net

Video Lesson on Composite Figures-Youtube. (March 24, 2009). [On-line Video]. Retrieved on May 25, 2009, from []

Composite Figures Title. (June 3, 2009)

Composite Figures Powerpoint Lesson. (June 4, 2009)