Sam Loyd's Missing Square/Variant

Puzzle
You have a square which is made from $4$ large triangles, $4$ small triangles, $4$ irregular octagons and $4$ small squares.

You jumble them up and reassemble the pieces once again into that same large square, but this time there is a hole in the middle.


 * MissingSquareVariant.png

Where did that hole come from?

Solution
The original shape that was made from the smaller shapes is not actually a square.

Each of the larger triangles has legs which are $3$ and $8$ units.

Hence it has area $12$ square units.

Each of the smaller triangles has legs which are $2$ and $5$ units.

Hence it has area $5$ square units.

Each of the small squares has sides which are $2$ units.

Hence it has area $4$ square units.

Each of the octagons has area $27$ units, which is determined by counting and measuring.

Hence the total area $\AA$ is given by:

The side of the large square, if it were a square, would be the hypotenuse of a right triangle whose legs are $13$ and $5$ units.

Hence by Pythagoras's Theorem, the side of that large square is $\sqrt {194}$ units.

That is, its area is $194$ square units.

So the shapes do not completely fill the large square.

That is because the hypotenuses of the large triangle and small triangle, when assembled into this shape, do not make a straight line.