Put another way, you cannot educate a child solely by teaching her academic subjects such as math and science. There's more to education than that. That's where sports and extracurricular activities come in. They allow your child to grow. They allow her to be stimulated by new ideas and ways of thinking. They give her a sense of accomplishment and encourage team spirit. These are all excellent lessons she needs to learn in order to be a successful adult.
Let's look at a couple of schools to get an idea of the rich array of choices your child will encounter at private school. Incidentally, it doesn't matter too much whether the private school you are considering is a day school or a boarding school. Either type of private school will offer more extracurricular activities
"A gifted person is someone who shows, or has the potential for showing, an exceptional level of performance in one or more areas of expression." ....National Association of Gifted Children
then it becomes obvious we have a huge responsibility on our hands. Because you have enriched your child's learning since birth, you understandably are anxious about placing him in a school setting where his giftedness will be misunderstood by both teachers and his peers. You want him to blossom and flourish in a formal school setting. You don't want limits placed on how fast he advances in a particular subject. You also want him to socialize in as normal a manner as possible. What then, besides home schooling, are your options?
The reality is that your choices for educating your gifted child may be limited by where you live and the programs available in your local public and private schools. Private schools for gifted children are highly specialized schools with well-qualified faculty who understand how to teach gifted children. Admission is generally based on several criteria one of which is an IQ test with a minimum of 125-140 the norm. The schools will also want to interview you and your child in order to see if there is a good match between your needs and requirements