Thursday, March 8, 2012

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

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