Finding Rectangular Dimensions Given Fixed Areas

Here is an excellent video made by one of your classmates to help you understand how to find whole number side lengths of possible rectangles given a fixed area.  Remember, we use the word 'fixed' here to mean constant or "stays the same."

Fixed Area

By: MJ & VM

Fixed Area is where you find different lengths and width(using whole numbers) of rectangles with a certain area. When solving a problem about finding area, simply find factors of the area. The U turn method could be very helpful when finding factors. YOU WILL HAVE TO FIND ALL POSSIBLE DIMENSIONS! Also, perimeter of the rectangle is required. You must find the length and width before finding the perimeter.  A fixed area with dimensions that is the most square like will have the smallest perimeter. The fixed area with dimensions that is the least square like will have the biggest perimeter. I believe this works just because of the squares and rectangles property’s.  

Here’s how a problem would look like step by step:

Find the all the possibilities of rectangles with a fixed area of 8cm2.

1. First, find each possible length and width by finding  factors of eight using the U turn method.

 U Turn Method  

1 * 8

2 * 4

4 * 2

8 * 1

MAKE SURE YOU ALWAYS USE THE OPPOSITE OF EACH RECTANGLE!!!!!!
(If a rectangle is 1 by 8, its opposite would be 8 by 1)

2. After you found all of the fixed area’s factors and its opposites, copy them into the length and width columns in a chart similar to the one below.

Chart

Length	Width	Perimeter	Area
1cm	8cm		8cm2
2cm	4cm		8cm2
4cm	2cm		8cm2
8cm	1cm		8cm2

ALWAYS LABEL YOUR MEASUREMENTS!

3. Once you have the length and width copied into the chart, find the perimeter of each rectangle. The perimeter can be determined by using the formula,2(L+W). Imagining the rectangle could help when finding perimeter.

Length	Width	Perimeter	Area
1cm	8cm	18cm	8cm2
2cm	4cm	12cm	8cm2
4cm	2cm	12cm	8cm2
8cm	1cm	18cm	8cm2

Notice how the perimeter changes. The perimeter of a shape will always stay the same but you can change the area.

Perimeter has to be involved in a fixed area problem because you need both the area and perimeter to describe size. One doesn't give enough information.

STILL REMEMBER TO LABEL PERIMETER AS WELL AS LENGTH, WIDTH, AND AREA!

4. Always check over your work in case you’ve made a mistake.

Partial Review Document for Covering & Surrounding

The following document is a partial review for the Covering & Surrounding unit.  You will also be responsible to understand surface area and volume of rectangular and triangular prisms.


Covering & Surrounding Partial Review

Creating Nets of Rectangular and Triangular Prisms

Creating nets can be challenging, but if you take some time to think about the number of sides of your prism and the which lengths meet, should be able to take the three dimensional prism to create a the two dimensional net.  The video video gives a clear presentation of this.

Area of an Irregular Rectangular Figure

Finding Area for

An Irregular Rectangular Shape

When you first take a look at the irregular rectangular shape below, you might be unsure how to find the area for the modified shape. However, finding the area for an irregular shape like the one below takes some steps in a process, but there are two different techniques to use to help find the area. First, let's take a glance at our irregular shape below.

IMPORTANT PRECAUTIONS: When you see the irregular shape, be sure to label the missing measurements so you can go on with either of the two techniques. For the width, we know that one part of the width is 5 units, and another unlabeled width line. To find the length of this line, we must subtract 7-5. So that answer is two units. So that line is 2 units. For the length, one part of the length on the right is 3 units. So we must subtract 5-3 to find the length. So that line is 2 units long. Now, our irregular shape has all the lines labeled.  This can be done because opposite sides of a rectangle are equal.  Since this shape can fit inside a rectangle and all lines are horizontal and vertical and create right angles, you can determine missing side lengths by subtracting the known lengths.

So now that all the lines are labeled, it's time to use either of the two ways to find our area.

WAY #1:   The first way to find the area of an irregular shape is to divide it into multiple rectangles. This will make finding the area much simpler, since you will be finding the area of each divided rectangle and adding up all the areas of each of those rectangles to find the total area.

 

Now that the shape has been divided into smaller rectangles, let's find the area of each of the rectangles. For Shape 1: 5 X 5 = 25 units squared. For Shape 2: 2 X 3 = 6 units squared. So 25 + 6 units squared gives us our total area of 31 square units.  Be sure you are using the appropriate lengths when you are finding area.  Notice to find the area of Rectangle 1 we are multiplying 5 X 5 not 5 X 7 even though the length of the whole side is 7 units.  That length is not the length of the rectangle we broke out of our entire figure.

Now, for the second way to find the area.

WAY #2:  The other way is to make the modified shape appear as a rectangle. So, let's go ahead and make the shape into an actual rectangle.

So now, we multiply 7 X 5 to get 35 units squared for our full rectangle. After that, we will now find the area of the rectangle that was the empty space. Since we know the length and width of the rectangle, we multiply 2 X 2 to get 4 units squared. Now, we subtract the area of the full rectangle from the empty space that was made a rectangle.  35 - 4 = 31 square units, which was the answer we got from the other way to fin d the area! Using either of these processes is a matter of preference, and both will work.

JUST BE SURE TO:

• TO REMEMBER TO LABEL ALL THE LINES IN THE SHAPE BEFORE PROCEEDING WITH THESE STEPS.
• MAKE SURE YOU KNOW THE AREA OF THE DIVIDED RECTANGLE YOU ARE ADDING/SUBTRACTING.

Hope you enjoyed the article (and found it helpful) on finding the area for the irregular rectangular shape.

Post Written by: RC

Fixed Area

Definition- A fixed area is when you are given a number and that is your defined area.  That area stays the same for all rectangles with whole number dimensions.

       

When finding the area of a rectangle you use the formula A=bh or Area = base * height.  A fixed area is when you are given a number as the defined area. You then need to find the base and height that would give you that area. When solving for a fixed area of a rectangle you are finding factors of the fixed area to determine the dimensions of the rectangles. The factors you are looking for would be the base and height of the rectangle that, when multiplied together, give you the fixed area. Drawing a table like the one below can help you solve the problem:

Base	Height	Perimeter	Area
1 in.	16 in.	34 in.	16 square in.
2 in.	8 in.	20 in.	16 square in.
4 in.	4 in.	16 in.	16 square in.

You can see the area stays the same above and that is why it is called "fixed."

        Filling the table (above) is easy. First, you make the table and add the columns and rows. Don't forget to label the columns (Base, Height, Perimeter, and Area). Next, find the factors for the fixed area and insert these into the base and height columns in pairs. Also, you must find the perimeter for a rectangle that has the base and height of the factors in the first two columns.  For example, the rectangle on the first line above has a base of 1 inch and a height of 16 inches, which are two factors of 16 sq. in, the fixed area in this case.  The perimeter of that rectangle would be 16+16+1+1 or 34 inches. Lastly, you have the area column to fill in, and obviously it's the same every row if there is a fixed area.

         While making the table, remember to include the unit of measurement.  If the area is given in square inches, then the base and height and perimeter will be in inches.  If you look at the table above you, can see all the units are inches. Only the area's unit of measurement is squared.

       

        Here are some other examples to prove that finding factors is a trick for fixed area.

        You can see that 12 and 3 are factors of 36. When you multiply them you get 36. Here is a factor rainbow that shows the common factors of 12.

        Since we have the factors of 12, we will use them to build a similar table with a fixed area of 12 square inches.  First fill in the area (in red). Then use the common factors in the factor rainbow to fill in the base and height (in blue).  Lastly, add 2b+2h to get the perimeter.

 

Base	Height	Perimeter	Area
1 in.	12 in.	26 in.	12 square in.
2 in.	6 in.	16 in.	12 square in.
3 in.	4 in.	14 in.	12 square in.

         In summary, using factors as the base and height for a given fixed area will allow you to master this area of math. 

Blog Post Written By: ES

Volume of Rectangular and Triangular Prisms

How to Find the Volume of Rectangular and Triangular Prisms

Volume is the amount of 3-D space that a object takes up.  Volume is measured in cubic units, because volume is 3-D and a cube is a 2-D square with another dimension.  A rectangular prism is just what it sounds like, a prism made of rectangles.  Think of a cube, but rectangular faces, so a little stretched out.  A triangular prism is  almost like a rectangular prism, but it’s end faces are triangles.  If you look at the prisms below, you can see that there are three parts labelled h, b, and l or w.  That h, b, and l or w stands for height, base and length or width (height, width and length being the first 3 dimensions, and the dimensions of a cube).  So, volume is basically figuring out how many of those cubed units fit in a figure.

If area is figuring out how many squared units are in a figure, and the formula for finding area for a rectangle is length by width, and a rectangular prism is a rectangle with height, it is logical for volume=height*length*width to be the formula for a rectangular prism.  You may see this formula, but it is more likely that you will see v=hlw, the shortened version.  The same theory goes for triangular prisms.  If  the formula for area is a= 1/2 lw, add height to make it v= 1/2 hlw.  Below are so practice problems and videos, and a very short Voki.

 

 

The following video gives and excellent representation of how cubic units work to determine volume.  Be sure to watch it to get a clear understanding.

This video explains triangular prisms and provides a couple of examples.

Blog Post Created By: SS and DK

Circumference of a Circle

What is the Circumference? The circumference is the distance around a circle or you can think about it as the perimeter of the circle

How to Find the Circumference of Circles:

To find the circumference of circles, first you must find the diameter or radius. The diameter of a circle is a line that goes through the center of the circle, and the radius is half of that (the distance between the center point of the circle and any point on the outside of the circle). Once you have found the diameter, you multiply by pi, or 3.14.

Class Activity:

Remember in class we measured the diameter of seven different circles.  We then compared the diameter to the measured circumference.  We found that the circumference was 3 and a little more times the diameter.  This is the diameter's relationship to pi.  Therefore to find circumference of a circle, simply multiply the diameter (or the radius doubled) to pi.

Here is a video that you might find useful.

Blog Post Created By: MM & CW

         

           

  

The formula for finding the area of a circle is radius squared times pi or r2 x 3.14

    

How it works:
When you square the radius of a circle it looks like the figure below. The radius squared fits into the circle three and a little bit more times. Pi is 3.14 which is three and a little bit more.  Remember that the radius of a circle is a measurement from the center of the circle to one point on the circle.  There are an infinite number of radii in a circle.  It is also half the diameter of the circle.  Think of what "radiate" means when trying to remember the definition of radius.  Radiate means to emit from a center point.

                               

The radius squared project we did in class showed us this hands on. Remember you took the diameter of a circle and squared it to actually create a square that perfectly fits around the circle.  Therefore, the diameter of the circle and the length of the side of the square that surrounds it is the same length.  Then you split the side lengths in half to create radius squares.  You then had four radius squares that made up the diameter square.

SO IN A NUTSHELL.....A=3.14 * r2

Click on the link for a video on finding the area of a circle
http://video.about.com/math/How-to-Find-the-Area-of-a-Circle.htm

Blog Post Created by: RR & LP