Triangle Problem

Given a triangle, such as the one below, how many different ways can you think of to measure the area?

                                                          
 
 

  1. Rectangle Method: In this method we use a rectangular grid that is tangent to each of the vertices of our triangle to find the area. We do this by subtracting the areas of the shaded triangles from the area of the rectangle to get the area of our original triangle.
     

     

                                                      

    Area of rectangle: 



  2. Grid Method: One of the most basic ways to measure the area of a figure is to put a grid over the object, like so.

                                                
 

The next thing to do is count the number of full square units are inside the triangle. After that we need to estimate the rest of the parts of the triangle, both of which are illustrated below.

                                                

Counting the number of full square units inside the triangle. (There are
28 full square units inside this triangle.)

                                                

Counting the leftover pieces of the inside of the triangle and estimating
the number of full square units those pieces make up. There are
approximately 18 full square units made up by the leftover pieces inside
this triangle.

After one does that little bit of estimation (you can see how the pieces add up on the attached sketch) all there is to do is add up the total number of full square units to find the area of the triangle. There are 28 full square units inside the triangle and 18 square units made up by the leftover pieces so that gives us a total of 46 square units; therefore, the approximate area of the triangle is 46 sq. units.
 

  1. Pickís Formula: Pickís Formula uses gridding to measure the area of a figure, but instead of using squares it uses points inside the figure and points lying on the border of the figure. Below is a sketch of our triangle showing all the interior points and border points. Pickís Formula states: 

                                               

There are 45 interior points and 4 exterior points. Putting these values into Pickís Formula gives us .
 

  1. Surveyors Method: In this method you take the coordinates of a figureÖit doesnít have to be a triangleÖand list them in descending order according to the graph and repeat the first coordinate as the last in order. The order of our points would be:

Points

Down Products

Up Products

4 , 10

0 , 3

4*3=12

0*10=0

12 , 1

0*1=0

12*3=36

4 , 10

12*10=120

4*1=4

The down products are the products of the diagonally down digits and the up products are the products of the diagonally up digits. The Surveyors Method states  So if we substitute our numbers in we get .

Note: The rest of the methods call for placing the triangle in the Cartesian coordinate plane so we can give the vertices of the triangle (x,y) coordinates.

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  1. Heron's Formula: In this method for finding the area of the triangle we are going to use Heronís Formula,

where 
I used the grid from method one to assist in putting the triangle into the Cartesian plane as pictured below.

First things first, we need the lengths of the sides of our triangle. To do that we need to utilize the distance formula,. We will call the distance between points (4,10) and (12,1) a; (0,3) and (12, 1) b; and (0,3) and (4, 10) c. By applying the distance formula we get that  Since  the perimeter of our triangle we add a, b, and c and divide by two (). Now that we have S, a, b,
and c we can substitute these values into Heronís formula, .


 
 

  1. Law of Cosines:  In this method we will use the Law of Cosines, , where are the sides of our triangle andequals the angle between the sides a and b and the formula  for the area of a triangle.


We will usewhich we got from our computations in method two.

Solving the Law of Cosines formula forgives us. Substituting our numbers into this formula gives us . Making another substitution into our area formula () shows us that .
 
 

  1. Algebra: In this method we will use plain olí algebra to derive the area of our triangle. To do this we are going to need a few things. We will need the length of the base (letís just say that side b is the base of our triangle) and we will need the height of the triangle. We already know the length of side b, so now we need to derive the length of the altitude of our triangle.

We know that the altitude of our triangle is going to be perpendicular to side b and pass through point (4,10). We need the equation for the line containing side b because we can then use its slope to find the slope of a line perpendicular to it (perpendicular lines have slopes that are negative reciprocals of each other). From here we can find the equation of the line containing the altitude of our triangle. Once we have that, we set the equations for side b and the altitude equal to each other and solve for x. Then you plug x back into either equation to get the y coordinate for the point of intersection. Now that we have to two points of our altitude we can use the distance formula to find the length of the altitude.

After all of that we need only to substitute the length of side b and the length of the altitude into .
 
 

  1. Calculus:  In this method are going to use integration to find the area of our triangle. To do this we need to find the area between two curves (in this case we are going to be taking the area between two lines at two different places). Look at the sketch below to see the two different areas that we are going to be finding.



 

Equation for side a: 

Equation for side b: 

Equation for side c: 

As you can see from the above sketches we are going to be looking at this equation for the area of our triangle: 

.

Summary: In summary, this assignment has shown that there are many different ways to solve the same problem. There are probably more than eight different ways to find the area of a triangle, but the ones here are the most obvious to me.

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