Franz and the Cake Fitting

Franz, the wedding cake maker, had some fickle customers. The bride wanted a wedding cake with a square top layer measuring 100 cm x 100 cm. The groom, however, insisted that the top layer be circular. Although he agreed that the layer should be the same area as the square layer the bride wanted.

After some words between them, the soon-to-be-wed couple asked Franz to bake them a cake with a circular layer on top.

Franz placed this circular layer in the smallest square box that he could manage. To the first decimal place, what were the dimensions of this box in centimeters?

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