## After buying a Gerver Sofa from the Moving Sofa Company, Jack wants to buy a matching cocktail table from the same company. Most important for him is that the table can be pushed through his L-shaped corridor into the living room without having to be lifted from its table legs.

Unfortunately, the simple square model offered to him is too small for him, so he asks for a bigger model.

He is offered the new pentagonal model illustrated below:

Note, while the shape and size can be ordered individually, due to the production process, all edges of the pentagonal table have to have the same length.

Given optimal form and size, what is the biggest pentagonal cocktail table (in terms of area) that Jack can buy that still fits through his unit wide L-shaped corridor?

Give your answer rounded to 10 digits after the decimal point (if Jack had choosen the square model instead the answer would have been 1.0000000000).

### Jack is looking for the largest pentagonal table that can slide through his unit wide L-shaped corridor. This problem is a well-known variant of the more famous moving sofa problem which asks about the sofa of largest area that would fit around the L-shaped corner when only sliding is allowed.

If Jack would have bought a square table, the side length of this table would have been 1 (since the width of the corridor is one unit) and thus, its area would be 1×1=1.

Now, regarding the pentagonal table, the table that encompasses the largest area able to negotiate the corner of his unit width L shaped hallway would be a regular pentagon (a five-sided polygon in which all sides are of equal length and every internal angle is 108 degrees). This is due to symmetry, which permits rotation as if there is enough space to move one edge-on into the corner, there must be enough to rotate the whole around the corner.

The formula for computing the area of a regular pentagon given its side s is:

Area = (5/4)*s²/tan(π/5)

Since his hallway is 1 unit wide, the side s of this largest possible pentagon is the distance that makes an angle of 108 degrees (internal angle of regular pentagon) and half the width of the corridor ending at the corner. Using basic geometry, this side length equals to 1/(2sin(54)), because the sine of angle A is the opposite side (1/2 in this case) divided by the hypotenuse (which is the side of pentagon we are trying to find).

So, s=1/(2sin(54)).

Substituting this value in the formula for area we have:

Area = (5/4) * [1/(2sin(54))]² / tan(π/5)

After evaluating this expression, we will find that the area of the pentagonal table is approximately 1.7204774006 (rounded to ten digits after the decimal point).

So, the pentagon that fits through the corner of his unit width hallway would have approximately 72% more area than the largest square that fits. Therefore the largest pentagonal cocktail table that Jack can buy that still fits through his L-shaped corridor would have an area of 1.7204774006.

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