A square number is the result of multiplying a whole number by itself. The first square numbers are 1, 4, 9, 16, 25 and 36, because 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9 and so on. That many dots fill a square exactly, like 16 dots in a 4 by 4 square.
A square number of dots fills a square exactly. Growing from one square to the next adds an L of the next odd number.
Formal definition and notation
A square number is a number of the form n2 = n × n for a whole numbern. Read n2 as “n squared”. The square numbers are 0, 1, 4, 9, 16, 25, … (some books start at 1). The nth square number equals the sum of the first nodd numbers: n2 = 1 + 3 + 5 + ⋯ + (2n − 1).
Drag the corner to make squares: 1 by 1, 2 by 2, up to 10 by 10. When length equals width, the area is a square number. Can you make a square with area 20?
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Square Number examples, step by step
Example 1: Find a square number
Warm-up
What is 92?
92 means 9 × 9.
9 × 9 = 81.
Answer:92 = 81, so 81 is a square number.
Example 2: Is it a square number?
Core
Is 72 a square number?
Find squares near 72: 82 = 64 and 92 = 81.
No whole number lies between 8 and 9, so no whole number squared equals 72.
Answer: No. 72 lies between the square numbers 64 and 81.
Yes, in most modern textbooks: 0 × 0 = 0, so 0 is the square of the whole number 0. Some elementary books start the list at 1 because a square of 0 dots is hard to draw. Either way, the positive square numbers are 1, 4, 9, 16, 25 and so on.
How can you tell if a big number is a square number?
First check the last digit: squares end only in 0, 1, 4, 5, 6 or 9, so 2,738 is out. Then find the two nearby squares. For 2,025, 402 = 1,600 and 502 = 2,500; a number ending in 5 must come from a root ending in 5, and 452 = 2,025. So 2,025 is a square number.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 3.OA.D.9, CCSS 6.EE.A.1
Cite this entry
Effortless Math Team. “Square Number.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/square-number/
Flashcard
TermSquare Number1, 4, 9, 16, 25, …
DefinitionA whole number times itself. 16 is a square number because 4 × 4 = 16: 16 dots fill a 4 by 4 square.
NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5Odd NumberAn odd number is a whole number that cannot be split into two equal groups: dividing it by 2 always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7 or 9, such as 7, 23 and 409. Every odd number can be written as 2k + 1, one more than a multiple of 2.Number System · Grade 2, Grade 3, Grade 4ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Factor PairA factor pair is two whole numbers that multiply to give a certain number. The factor pairs of 12 are 1 × 12, 2 × 6 and 3 × 4. Listing factor pairs is a sure way to find every factor of a number: start at 1 and stop once the pairs begin to repeat.Number System · Grade 4, Grade 5, Grade 6MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2Perfect SquareA perfect square is a number that equals a whole number times itself, so its square root is a whole number. For example, 49 is a perfect square because 7 × 7 = 49 and √49 = 7. The first perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.Number System · Grade 6, Grade 7, Grade 8Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4ArrayAn array is an arrangement of objects in equal rows and equal columns, forming a rectangle. A 4 by 6 array has 4 rows of 6, so it holds 4 × 6 = 24 objects. Arrays show multiplication as equal groups, and turning one sideways shows that 4 × 6 = 6 × 4.Operations · Grade 2, Grade 3, Grade 4Pythagorean TheoremThe Pythagorean theorem says that in a right triangle, a2 + b2 = c2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Geometry · Grade 8, Algebra 1Multiplication TableA multiplication table is a grid that lists the products of pairs of numbers, usually 1 to 10 or 1 to 12. To read it, find one factor along the top row and the other down the left column; the box where they meet is the product. Row 6 meets column 7 at 42, since 6 × 7 = 42.Operations · Grade 2, Grade 3, Grade 4