Area puzzle in colored triangle

Question

I have tried to figure out by calculating the area but I got same results for these, so where is gone the hole?

Answer 1

This is a tikz code snippet for the drawing in $\LaTeX$ for everyone to grab and try it yourself:

\documentclass{article}
\usepackage{tikz,pgfplots}
\begin{document}
\tikzset{polygon/.style = {thick, line join=bevel, fill opacity=0.6}}
\newcommand{\redshape}[1]
{\draw [polygon, fill=red] #1 -- ++(8,0) -- ++(0,3) -- cycle;}
\newcommand{\tealshape}[1]
{\draw [polygon, fill=teal] #1 -- ++(5,0) -- ++(0,2) -- cycle;}
\newcommand{\yellowshape}[1]
{\draw [polygon, fill=yellow]#1 -- ++(2,0) -- ++(0,1) 
-- ++(3,0) -- ++(0,1) -- ++(-5,0) -- cycle;}
\newcommand{\greenshape}[1]
{\draw [polygon, fill=green]
#1 -- ++(5,0) -- ++(0,2) -- ++(-3,0)
-- ++(0,-1) -- ++(-2,0) -- cycle;}

\begin{tikzpicture}[scale=1]
\draw [gray](0,0) grid(13,11);

# FIRST SET
\redshape{(0,0)}
\yellowshape{(8,1)}
\greenshape{(8,0)}
\tealshape{(8,3)}

# SECOND SET
\redshape{(5,8)}
\yellowshape{(5,6)}
\greenshape{(8,6)}
\tealshape{(0,6)}

\end{tikzpicture}

Notice in the \newcommand part for the \redshape and \tealshape, the redshape drawing sequence starts with the left bottom corner, goes right for 8 units then goes up for 3 units (-- ++(8,0) -- ++(0,3) part), while the tealshape starts from the bottom left corner, goes right for 5 units and then goes up for 2 units(- ++(5,0) -- ++(0,2) part).

Once we defined all the shapes in the \newcommand part, we don't modify them, and just rearrange them in the \tikzpicture part.

These two shapes have different slopes, hence the big combined shape is not a triangle, you can't use the triangle formula for the big shape, and say, hey, there is a square missing! Instead, you have to add the area of each single shape: red is $8\times 3/2$, teal is $5\times 2/2$, yellow occupies 7 grids, and green occupies 8 grids, and you will find that the area is conserved.

Answer 2

When calculating the total area of those four shapes in square units, placement doesn't matter.

The perceived total area of the first triangle is $13*5/2 = 32.5$.

The perceived total area of the second triangle is one less, $31.5$.

The red triangle has area $3*8/2 = 12$. The green triangle has area $5*2/2 = 5$. The yellow polyomino has area $7$ and the green polyomino has area $8$. Thus the total area in both cases is $32$.

As was noted in the comments, the apparent discrepancy is caused by the two triangles not having the same slope, causing the diagonal to bend in or out slightly, while appearing at first glance to be relatively straight.