Square numbers 1 to 100: the list (1, 4, 9, 16, 25, 36, 49, 64, 81, 100), the squares from 1² to 30², the cube numbers to 10³, what a square number is and why it is called that, how to spot one, the patterns (odd numbers, last digits, differences), square roots, and a printable chart with practice

"Square numbers" is typed in the UK about 25,000 times a month, and "square numbers 1 to 100" another 2,200, almost all by Year 5 and 6 children and their parents, because the national curriculum asks children to "recognise and use square numbers and cube numbers, and the notation for squared (²) and cubed (³)" in Year 5, and the KS2 SATs and the 11-plus both test it. This page gives the list to 100, the table of squares to 30², the cube numbers, the reasons behind the name, the quick ways to spot a square, the patterns that make the list easy to remember, and the link to square roots and Pythagoras, with a printable chart and practice. Every square root also has its own page in our maths library.

Download the square and cube numbers chart with practice (PDF) (A4, the squares 1² to 100² and the cubes 1³ to 20³ on page 1, the patterns and the practice set with answers on page 2, free).

The square numbers from 1 to 100

1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

These ten are the ones to know by heart in Year 5 and Grade 4; they are the diagonal of the times-table grid. The next five, 121, 144, 169, 196 and 225 (11² to 15²), are expected by GCSE and in Grade 8, where 8.EE.2 asks for "square roots of small perfect squares".

What a square number is, and why "square"

A square number (a perfect square) is the product of a whole number and itself. 7 × 7 = 49, written 7² and read "seven squared" or "seven to the power of two". The little 2 is the index (or exponent, or power) and says how many sevens are multiplied.

The name is literal. Take 16 counters: they can be laid out as a 4-by-4 square with no counters left over. 15 counters cannot make a square of any size. So the square numbers are exactly the numbers of dots that fill a square, which is also why the area of a square with side 4 cm is 4² = 16 cm². Drawing the dot squares for 1, 4, 9, 16 and 25 is the Year 5 lesson that makes the idea stick.

The squares from 1² to 30²

The PDF carries the table on to 100² = 10,000. Useful landmarks: 20² = 400, 25² = 625, 30² = 900, 40² = 1,600, 50² = 2,500, 100² = 10,000. Multiples of 10 are easy: square the digit and add two zeros.

The cube numbers

A cube number is a whole number multiplied by itself twice: 3 × 3 × 3 = 27, written 3³, "three cubed". The name is literal again: 27 small cubes build a 3-by-3-by-3 cube.

GCSE (both tiers) expects 1³, 2³, 3³, 4³, 5³ and 10³ from memory. The numbers that are both square and cube are the sixth powers: 1, 64 (8² and 4³), 729 (27² and 9³).

How to tell whether a number is a square number

The last-digit rule. A square can only end in 0, 1, 4, 5, 6 or 9. It can never end in 2, 3, 7 or 8. So 1,234, 587, 3,333 and 98 are not squares, without any working. (Why: squaring the digits 0 to 9 gives 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, and only the last digit of a number affects the last digit of its square.) The rule rules numbers out, not in: 15 ends in 5 and is not a square.

The digit-sum rule. Add the digits until one digit remains (the digital root). A square's digital root is always 1, 4, 7 or 9. 2,025: 2 + 0 + 2 + 5 = 9, passes (and 45² = 2,025). 2,026: digital root 1, passes the test but is not square, so again the rule only rules out.

The sure test. Find the square root. If it is a whole number, the number is a square. For numbers under 1,000, know the table above; for bigger ones, estimate: √2,500 = 50, so √2,601 is a bit more, try 51: 51² = 2,601. Yes.

Prime factors. A number is a square exactly when every prime in its factorisation appears an even number of times. 144 = 2⁴ × 3²: even powers, square. 72 = 2³ × 3²: the 2 appears three times, not square.

The patterns that make the list easy

• Add the odd numbers. 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16. The nth square is the sum of the first n odd numbers. So the gaps between squares are the odd numbers: 3, 5, 7, 9, 11, ... (from 100 to 121 is 21, from 121 to 144 is 23).
• Next square from this one. (n + 1)² = n² + 2n + 1: 15² = 14² + 28 + 1 = 196 + 29 = 225. Children who know 10² to 20² can get 21² to 30² this way in their heads.
• Squares of 5s. A number ending in 5: multiply the tens digit by one more than itself and write 25 after it. 35² = (3 × 4)25 = 1,225. 85² = (8 × 9)25 = 7,225.
• Difference of two squares. a² − b² = (a + b)(a − b): 41² − 39² = 80 × 2 = 160. This is the GCSE identity, and it also explains why consecutive squares differ by an odd number.
• Every square is a multiple of 4 or one more than a multiple of 4. Even squares (4, 16, 36) divide by 4; odd squares (9, 25, 49) leave remainder 1. A number that leaves remainder 2 or 3 on dividing by 4 is never a square.
• Triangular numbers. Two consecutive triangular numbers (1, 3, 6, 10, 15, ...) add to a square: 6 + 10 = 16, 10 + 15 = 25.

Square roots: the inverse

The square root undoes squaring: since 9² = 81, √81 = 9. Every square number has a whole-number root, and every other number has a root that goes on for ever (√2 = 1.41421..., which is why √2 is called irrational). KS3 and Grade 8 expect the roots of the squares to 15² (and 20², 25², 30²) from memory, and estimation for the rest: √50 is between 7 and 8, nearer 7 (7.07). Strictly 81 has two square roots, 9 and −9, since (−9)² = 81 too; the √ sign means the positive one.

The place squares matter most after Year 6 is Pythagoras' theorem: in a right-angled triangle the square on the longest side equals the two other squares added. 3² + 4² = 9 + 16 = 25 = 5², so a 3-4-5 triangle is right-angled; so are 5-12-13 and 8-15-17 (every term a whole number, which is what makes these triples turn up in exams). Knowing the squares to 20² by heart turns a Pythagoras question from a calculator exercise into mental arithmetic.

Where it sits in each curriculum

Practice (answers in the PDF)

1. Write all the square numbers between 20 and 150.
2. Which of these are square numbers: 121, 128, 169, 180, 225, 250? Say how you know without a calculator.
3. What is 13² − 12²? Find it two ways.
4. Which number below 100 is both a square number and a cube number (other than 1)?
5. Work out 45² using the "ends in 5" trick, and check by long multiplication.
6. A square garden has an area of 196 m². How long is each side? What would the side be if the area were 200 m² (to one decimal place)?

Download the chart and the practice set (PDF)

Where to get help

A 1-on-1 online maths class with a tutor who holds a master's in mathematics and teaches KS2, KS3 and GCSE, the Australian Years 5 to 9, CBSE Class 5 to 8 and US Grades 4 to 8 every week: the squares learned as dot pictures and odd-number sums first, then as facts to recall, then put to work in roots and Pythagoras. The maths library has a page for every square-root question children type (and the square roots 1 to 30 table); the free 30-minute first class is a real lesson on this week's homework. The partner page is prime numbers to 100.

Questions parents ask

See how we teach this, 1-on-1 online →