Teaser 2978: Norfolk Flats

From The Sunday Times, 20th October 2019 [link]

Sam has purchased Norfolk Flats; an area of farmland (less than 100 hectares) bordered by six straight fences. He intends to farm an area which is an equilateral triangle with corners that are the midpoints of three of the existing boundaries. This creates three more distinct areas (one for each of his sons); these areas are identical in shape and size and have two sides that are parallel.

Sam measured the area (in square metres) which each son will farm and also his own area. One of the numbers is a square and the other a cube. If I told you which was which, you should be able to work out the area of Norfolk Flats.

What (in square metres) is that area?

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