Wave Project

Demonstration of the Area
 formula for Parallelograms

Clear[tall,wide,grid]
poly=Graphics[{RGBColor[0,1,0],Thickness[.02],
             Line[{{2,3},{6,3},{4,0},{0,0},{2,3}}]}];

grid[tall_,wide_]:= Table[
		Graphics[{RGBColor[0,1,1],{Line[{{a,tall},{a,0}}]},
		Line[{{0,b},{wide,b}}]}],{a,0,wide},{b,0,tall}];

cut=Graphics[{RGBColor[1,0,0],Line[{{2,3},{2,0}}]}];

Show[grid[5,6],poly,cut,
Graphics[Text["base = 4 units",{2.1,0.3},{-1,0}]],
Graphics[{RGBColor[1,0,0],Text["height = 3 units",{2.1,2},{-1,0}]}],
	Axes->True];

Consider the above parallelogram.
Watch what happens as we cut off the triangle shape off of the left of the parallelogram and move it right.

tri[d_]:=Polygon[{{0+d,0},{2+d,3},{2+d,0}}];

trap=Graphics[{RGBColor[0,1,0],Thickness[0.01],
      Line[{{2,0},{4,0},{6,3},{2,3},{2,0}}]}];
      
Do[Show[grid[3,6],trap,Graphics[{RGBColor[1,0,0],tri[d],Thickness[.1]}],grid[4,6],
Axes->True,AspectRatio->Automatic],{d,0,4}]

Clear[tri,trap]

poly=Graphics[{RGBColor[0,1,0],Thickness[.02],
              Line[{{2,2},{9,2},{7,0},{0,0},{2,2}}]}];

cut=Graphics[{RGBColor[1,0,0],Line[{{2,2},{2,0}}]}];

Show[poly,grid[3,10],cut,
     Graphics[Text["base = 7 units, height = 2 units",
             {2,3},{-1,0}]],
     Axes->True,
AspectRatio->Automatic];

Consider the above parallelogram.
   Area = 14 units  (count the squares inside!)
              
What happens when we cut the triangle shape off of the left of the parallelogram and move it right?

tri[d_]:=Polygon[{{0+d,0},{2+d,2},{2+d,0}}]

trap=Graphics[{RGBColor[0,1,0],Thickness[.01],
              Line[{{2,2},{9,2},{7,0},{2,0},{2,2}}]}];
              
Do[Show[trap,Graphics[{tri[d],Thickness[.1]}],grid[3,10],
        Axes->True,AspectRatio->Automatic, 
        PlotRange->{{0.2,10},{-0.2,2.2}}],
   {d,0,7}]

Therefore,  
Area of Parallelogram = Area of Rectangle
           A = 7(2) or A=14

Let's change the shape of the parallelogram.

poly2=Graphics[{RGBColor[0,1,0],
                  Polygon[{{1,8},{3,8},{2,0},{0,0}}]}];
                  
Show[poly2,grid[8,3],Axes->True,AspectRatio->Automatic,
PlotRange->{{0,3},{0,8}}];

This parallelogram has a base = 2, height = 8.

trap2=Graphics[{RGBColor[0,1,0],
               Polygon[{{1,8},{3,8},{2,0},{1,0}}]}];
               
cut=Graphics[{RGBColor[1,0,0],Polygon[{{0,0},{1,0},{1,8}}]}];

Show[trap2,cut,Axes->True,AspectRatio->Automatic,
PlotRange->{{-0.2,3.2},{-0.2,8.2}}];

tri[d_]:=Polygon[{{0+d,0},{1+d,0},{1+d,8}}];

Do[Show[trap2,Graphics[{RGBColor[1,0,0],tri[d]}],
        Axes->True,AspectRatio->Automatic,
        PlotRange->{{-0.2,3.2},{-0.2,8.2}}],
   {d,0,2}];

Area of a Parallelogram = Area of a Rectangle
                        A = bh

Let's consider a parallelogram that has a base of 4 and a height of 7.

poly3=Graphics[{RGBColor[0,0,1],
                  Polygon[{{1,0},{9,7},{13,7},{5,0}}]}];
                  
cut=Graphics[{RGBColor[1,0,0],Line[{{9,7},{9,0}}]}];
                  
Show[poly3,grid[8,14],cut,
Graphics[Text["base = 4",{5.5,0.3},{-1,0}]],Graphics[{RGBColor[1,0,0],Text["height = 7",{5,4},{1,0}]}],
Axes->True,AspectRatio->1,PlotRange->{{0,15},{0,8}}];

Now let's cut this parallelogram into shapes.

cut2=Graphics[{RGBColor[1,0,0],Line[{{5,3.5},{5,0}}]}];

Show[poly3,cut,cut2,Graphics[{Text["SHAPE 1",{5.2,3},{-1,0}],
Text["SHAPE 2",{2.2,1}, {-1,0}]}],
Axes->True,AspectRatio->1,PlotRange->{{0,15},{0,8}}];

Let's move the parallelogram( SHAPE 1 ).

Clear[tri]

shape1=Graphics[{RGBColor[1,0,0],
           Polygon[{{9,3.5},{9,7},{13,7}}]}];

parr[d_]:=Polygon[{{5+d,0},{9+d,3.5},{9+d,7},{5+d,3.5}}];

tri[c_]:=Polygon[{{1+c,0},{5+c,0},{5+c,3.5}}];

Do[Show[shape1,Graphics[{RGBColor[0,0,1],parr[d]}],
        Axes->True,AspectRatio->Automatic,
        PlotRange->{{0,14},{0,8}}],
   {d,0,4}];

Let's move the triangle ( SHAPE 2).

shape2=Graphics[{RGBColor[1,0,0],
           Polygon[{{9,0},{9,7},{13,7},{13,3.5}}]}];
           
Do[Show[shape2,Graphics[{RGBColor[0,0,1],tri[c]}],
        Axes->True,AspectRatio->Automatic,
        PlotRange->{{0,14},{0,8}}],
   {c,0,8}];

Base is still 4 units, height is 7 units.  Area=base x height
A = bh or A = 4(7) = 28 square units.