One square kilometre is 0.3861021585 square miles. As with every areal conversion the number is the linear factor squared: a mile is 1.609344 km, and 1.609344² is 2.58999, whose reciprocal gives the figure above.
About the units
The square kilometre is the standard unit for describing the area of countries, regions, lakes and protected land almost everywhere. The square mile is its counterpart in the United States and, less consistently, the United Kingdom. The square mile has one property the square kilometre lacks: it is the basis of the American Public Land Survey System, where a section is exactly one square mile of 640 acres, a township is 36 sections, and the grid those definitions produced is visible from the air across the American Midwest as a pattern of square fields and right-angled roads.
The exact factor
One square mile is exactly 2.589988110336 km², since 1.609344² = 2.589988110336. The reciprocal, 0.386102158542446, is the figure for the other direction. The rough approximation is that a square mile is about two and a half square kilometres, or that a square kilometre is a bit under four tenths of a square mile. For scale: Greater London is about 1,572 km² or 607 mi²; the state of Texas is 695,662 km² or 268,596 mi².
Where you meet this conversion
Geography, demography and news reporting. Population density is the case where the conversion matters most and is most often botched, because the unit appears in the denominator: converting 100 people per square kilometre to people per square mile means multiplying by 2.59, not dividing. Wildfire extent, flood zones, national park sizes and oil spill coverage are all reported in one unit inside the United States and the other internationally. Military and aviation contexts sometimes add a third unit, the square nautical mile, which is 3.4299 km².