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Is It a Leap Year? How to Calculate Leap Years

Updated on February 10, 2015
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TR Smith is a product designer and former teacher who uses math in her work every day.

What Are Leap Years and Common Years?

Leap years are years which have an extra day in the calendar, 366 instead of the usual 365. The extra day is February 29, often called Leap Day. All other years are called common years.

What is the Point of Leap Year?

Why do we add an extra day to some years? The reason is because the length of our calendar year does not precisely coincide with the length of a tropical year (or solar year) in astronomy. A calendar year is defined as the length of time between January 1 and December 31, usually 365 days in the Gregorian calendar we currently use. In astronomy, a tropical year is the amount of time it takes the sun to return to the its starting position in our seasonal cycle, about 365.2421897 days, where a day is defined as 86400 seconds.

Since the sun does not take exactly 365 days to return to position, over time, our 365-day calendar would become badly out of sync with the astronomical solar year. To re-sync the years, we add an extra day to the calendar every few years. This ensures that the vernal equinox is always around March 21 each year.

When Is Leap Year?

Leap years occur during years that are multiples of 4, but not multiples of 100, unless the year happens to be a multiple of 400. Non-leap years are called common years. For instance,

  • 2011 is a common year with 365 days since the number 2011 is not a multiple of 4.
  • 2040 will be a leap year with 366 days, since 2040 is divisible by 4 and not divisible by 100.
  • 2100 will be a common year since although the number 2100 is divisible by 4, it is also divisible by 100 and not divisible by 400.
  • 2400 will be a leap year since 2400 is a multiple of 4 and a multiple of 400.

The cycle of leap years repeats every 400 years. For every 400 consecutive years, 97 of the years will be leap years and 303 will be common years. This means the average number of days in a year under the Gregorian calendar is calculated as the weighted average

[303*365 + 97*366]/400 = 365.2425

which is very close to the length of a solar year.

Leap Year Algorithm

The mathematical algorithm to determine whether a year is a leap year or common year is as follows.

  • (1) Check if the year is divisible by 4. If it is not divisible by 4, then it is a common year. Otherwise go to the next step.
  • (2) Check if the year is divisible by 100. If it is not divisible by 100, then it is a leap year. Otherwise, go to the next step.
  • (3) Check if the year is divisible by 400. If it is not divisible by 400, then it is a common year. If it is divisible by 400, it is a leap year.

In summary, a year is a leap year if it is divisible by 4, except for the years that are divisible by 100 by not divisible by 400. For example, 1888 was a leap year, 1900 was a common year, and 2000 was a leap year. To check if a year is leap or common, you can use the calculator below.

Leap Year Calculator

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Recycling Calendars: When Does a Calendar Repeat?

Without leap years, 365-day calendars would repeat every 7 years. But with leap years, the maximum length of time for a calendar to repeat is 28 years. Common years repeat more often within the 28 year cycle, for example, the years

1901, 1907, 1918, 1929, 1935, 1946, 1957, 1963, 1974, 1985, 1991, 2002, 2013, 2019,...

all have the same calendar. Leap years repeat exactly once every 28 years. For example, the calendar for 1988 is the same as for 2016. This means you can recycle old calendars and date books for future use.

An Alternative Leap Year System: 128-Year Cycles

The purpose of leap year is to make the average length of a year as close to 365.2421897 days as possible. The Gregorian calendar uses a 400-year cycle with an average year length of 365.2425 days, which is only 0.000085% longer than the tropical year. However, you can devise other leap year systems that are even closer.

Suppose you make Leap Day occur in years that are divisible by 4 but not divisible by 128. Then you would get a 128-year calendar cycle with 31 leap years and 97 common years. The average length of a year would be given by the weighted average

[97*365 + 31*366]/128 = 365.2421875

which is even closer to the length of a solar year.

The exact length of one second is based on physical properties of cesium 133, the only stable isotope of cesium. This standard was set in 1967.

Leap Seconds

Occasionally, one extra second is added to the end of June 30 or December 31 in Coordinated Universal Time (UTC), the global time standard. This is done to correct irregularities in the Earth's rotation, since otherwise we would slowly drift away from International Atomic Time, another clock standard. These extra seconds are called leap seconds. Since 1975, 25 leap seconds have been added. The most recent additions were

  • June 30, 2015
  • June 30, 2012
  • December 31, 2008
  • December 31, 2005
  • December 31, 1998
  • June 30, 1997

There is no set pattern for when the leap seconds are added. The insertion of leap seconds is decided by the International Earth Rotation and Reference Systems Service (IERS), whose current policy is to add a leap second whenever the difference between UTC and UT1 (another variant of universal time) approaches 0.6 seconds.

Leap Day Fun Facts: February 29 in History and Pop Culture

  • Sir James Wilson, former Premier of Tasmania, was born February 29, 1812 and died February 29, 1880.
  • In Ireland, Bachelor's Day is February 29. On Bachelor's day women can propose to men, and according to tradition if the man declines he must buy the woman a gown, fur coat, or 12 pairs of gloves.
  • Some cultures consider it unlucky to be born or to wed on Leap Day.
  • People born on Leap Day are called leaplings.


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