Check the laws in your state before determining how many days to dedicate to schooling each year. Laws vary widely from state-to-state, from absolutely no minimum attendance requirements, to detailed regulations as to how many hours per day and days per year. Bridgeway Academy's website has the homeschooling laws of all 50 states in handy chart form.
The traditional school year is nine months, usually September through May. After subtracting weekends, holidays and teacher in-service days, there are 180 days of school per year, but this is not the standard across all states.
The reason for the three month summer break, usually thought to be related to the need for harvest labor on farms, is largely obsolete. If your state allows, you could spread your school days over the entire year. But arranging your calendar in sync with your local district's has some advantages. For example, your kids will be in school when everyone else is, so coordination with their friends is easier.
Also, you might find that you as a teacher can use a summer break even more than your students. It takes a while to plan the next year and it's easier when you're not teaching at the same time. Summer trip plans and the time needed to tend a large garden might also influence your summer break decision.
Schooling year round does leave more flexibility for holidays than the traditional school calendar, say for ski vacations, or an extended Christmas visit with grandparents.
State laws also dictate when children must begin to be educated. Home-schoolers in your state may or may not be subject to these laws. The usual age is 6, but it might be 5, or as old as 8.
If your child is younger than the compulsory age, you can relax your days-per-year plan dramatically. Young children learn wonderfully through just living and playing. If you make available a rich learning environment, a little intentional seizing of teachable moments will provide all the education they need.
The requirement for compulsory education might expire at 16 years old in your state, freeing you to be more creative in your yearly calendar and what direction your teen's education might take at that age.
Some curricula may take longer than nine months to finish. If you have the freedom to do so, you might consider announcing that math is over for the year when the book is finished. This can be highly motivating for some students.
If you enroll your child in a correspondence school, the yearly calendar will be set for you.