It's... not? It's 9.87. If you throw a dart at the number line, the average distance it will be from the nearest integer is 0.25. The distance from pi^2 to 10 is 0.13. That's below average, but in the second quintile. There's nothing special about 10, other than it happens to be the number of fingers on two human hands.

However, deriving approximations for pi from the Basel series is sort of interesting. Except summing the Basel series requires at a bare minimum the theory of Taylor series, so it is not an accessible theorem for a primitive geometer.