Problem 667

Moving Pentagon

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 chosen the square model instead the answer would have been 1.0000000000).

移动的五边形

在从移动沙发公司购买了一套格弗式沙发之后，杰克打算再买一张配套的鸡尾酒桌。对他来说最重要的是，这张桌子能够直接推着通过他家的L形走廊而不需要被抬起来。
很不凑巧，那些能够满足杰克的条件的正方形桌子都太小了，他想要再大一些的桌子，于是销售人员向他推荐了一些新的五边形桌子，如下所示：

需要注意的是，尽管桌子的形状和大小都可以定制，但是由于制造工艺的限制，五边形桌子的每一条边必须有相同的长度。

在最优的形状和大小下，杰克能够买到的、能顺利通过他家的L型走廊的五边形鸡尾酒桌到底能有多大（以桌面的面积计）？

将你的答案保留小数点后10位小数（比方说，如果杰克打算买一开始那张正方形桌子，那么答案就应该是1.0000000000）。