# The Commonwealth of Poland and Lithuania

The latest edition of In Our Time on Radio 4 (first broadcast on 14 October 2021), the programme that tries to deal in depth with a narrow topic, was about the medieval elective kingdom of Lithuania and Poland (or the other way round).

One of the speakers, as he introduced the topic, described the kingdom as covering an area of “one million kilometres squared”.

Has the difference between “square kilometres” and “kilometres squared” changed since I was at school? (or similarly square miles and miles squared).

To me the phrases “one kilometre square(d)” and “one square kilometre” indicate identical areas, but for any larger amount, the difference between the two will only increase as the numbers get bigger. So, “two kilometres square(d)” is equal to “four square kilometres” (because “2 kilometres square(d)” is equivalent to saying “an area which stretches 2 kilometres along the x axis and 2 kilometres along the y axis”). By my calculation, using this logic, “1 million kilometres squared” is the same as “1 million million square kilometres”. The land surface area of the earth is said to be about 150,000,000 sq km according to Science Desk Reference American Scientific, so presumably the expert speaking this morning meant to say “one million square kilometres”. If an expert can get a figure wrong by this magnitude (which makes the area of Poland and Lithuania about 6,660 times bigger than the total land area of the earth), what hope have we for the rest of the population?

Unless, of course, the meanings of “square kilometres” and “kilometres square(d)” are now identical. In which case, how do we know how to convert an areal measure in miles to its equivalent in kilometres? (4 square miles is 2 miles in each direction, but as 2 miles is equivalent to ca 3.2 kilometres, 4 square miles is 10.24 square kilometres, while 4 miles squared is 16 square miles, or 40.96 square kilometres.)

If these two expressions have now coalesced, then the language has lost a valuable distinction, and may well cause confusion to budding mathematicians, to say nothing of architects and land surveyors.

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