225 (two hundred ndtwenty-five) is the natural number following

224 Year 224 ( CCXXIV) was a leap year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Iulianus and Crispinus (or, less frequently, year 977 ''Ab urbe con ...

and preceding

226 Year 226 ( CCXXVI) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Marcellus (or, less frequently, year 979 ''Ab urbe cond ...

In mathematics

225 is the smallest number that is a polygonal number in five different ways. It is a

square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The u ...

, an octagonal number, and a

squared triangular number In number theory, the sum of the first cubes is the square of the th triangular number. That is, :1^3+2^3+3^3+\cdots+n^3 = \left(1+2+3+\cdots+n\right)^2. The same equation may be written more compactly using the mathematical notation for summa ...

. As the square of a

double factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the ...

, counts the number of

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...

s of six items in which all cycles have even length, or the number of permutations in which all cycles have odd length. And as one of the

Stirling numbers of the first kind In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed poin ...

, it counts the number of permutations of six items with exactly three cycles. 225 is a highly composite odd number, meaning that it has more divisors than any smaller odd numbers. After 1 and 9, 225 is the third smallest number ''n'' for which , where ''σ'' is the sum of divisors function and ''φ'' is

Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...

. 225 is a

refactorable number A refactorable number or tau number is an integer ''n'' that is divisible by the count of its divisors, or to put it algebraically, ''n'' is such that \tau(n)\mid n. The first few refactorable numbers are listed in as : 1, 2, 8, 9, 12, 18, ...

. 225 is the smallest

to have one of every digit in some number base (225 is 3201 in base 4) 225 is the first odd number with exactly 9 divisors. 225° is five eighths of a circle.

In other fields

* The years AD 225 and

225 BC __NOTOC__ Year 225 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Papus and Regulus (or, less frequently, year 529 ''Ab urbe condita''). The denomination 225 BC for this year has been u ...

.225 Winchester The .225 Winchester cartridge was introduced in 1964 by the Winchester Repeating Arms Company The Winchester Repeating Arms Company was a prominent American manufacturer of repeating firearms and ammunition. The firm was established in 186 ...

, firearm cartridge * 225 is a track from

New Model Army The New Model Army was a standing army formed in 1645 by the Parliamentarians during the First English Civil War, then disbanded after the Stuart Restoration in 1660. It differed from other armies employed in the 1639 to 1653 Wars of the Th ...

album ''

Thunder and Consolation ''Thunder and Consolation'' is the fourth studio album by English rock band New Model Army, released on 15 March 1989 by EMI Records. The album stands as a landmark in the New Model Army catalogue, being their most successful album to date and ...

'' * 225 squares on a scrabble board

References

Integers {{Num-stub