Maths
What are square numbers?
Quick answer
A square number is what you get when you multiply a whole number by itself, like 4 × 4 = 16. The first square numbers are 1, 4, 9, 16, 25 and 36.
In one sentence
A square number is the result of multiplying a whole number by itself, like 4 × 4 = 16, and you can always arrange that many dots into a perfect square.
Explained
Lay out tiles in a square: 3 rows of 3 uses 9 tiles. 4 rows of 4 uses 16. Those totals, 9 and 16, are square numbers. Try 10 tiles and you can't make a square without gaps or spares, so 10 is not a square number.
window
We write squares with a small raised 2: 5² means 5 × 5 = 25. Say it "five squared". The little 2 is called a power or index (see powers and indices).
The list of square numbers
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
They are hiding in your times tables grid, running diagonally from the top-left corner. Zero counts too (0 × 0 = 0), though school lists usually start at 1.
Patterns in square numbers
- Odd-number gaps: 1, 4, 9, 16, 25 go up by 3, 5, 7, 9, the odd numbers in order. Each time you grow the square by one row and one column, you add the next odd number of tiles.
- Last digits: a square number always ends in 0, 1, 4, 5, 6 or 9, never 2, 3, 7 or 8. So 47 and 88 can't be square.
- Factors: square numbers have an odd number of factors. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36, nine in all, because 6 pairs with itself.
Squares, area and square roots
Squares link straight to area: a square garden with sides of 6 m has an area of 6² = 36 m². Going backward, from 36 to 6, is finding a square root.
Examples
- 1² = 1
- 9² = 81
- 15² = 225
- 20² = 400 (2² = 4, then add two zeros)
- 0.5² = 0.25
- (−3)² = 9, since a negative times a negative is positive
Common mistakes
- Doubling instead of squaring. 6² is 36, not 12.
- Thinking 1 doesn't count. 1 × 1 = 1, so 1 is the first square number.
- Squaring only part of a number. 30² is 900, not 90. Fix: 30 × 30 = 3 × 3 × 10 × 10.
Quick quiz
Square numbers
Check what you know with 4 quick questions.
Question 1 of 4
What is 7 squared (7²)?
Practise it
Try it
- What is 8²?
- Is 50 a square number?
- Which of 27, 36 and 48 is a square number?
- A square rug has an area of 81 m². How long is each side?
Answers: 1) 64. 2) No: 7² = 49 and 8² = 64, so 50 falls between them. 3) 36 = 6². 4) 9 m, because 9 × 9 = 81.
Squares, roots and powers practice
Practise it step by step in a full course: Grade 7 Maths
For grown-ups
Related topics
Square roots
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 25 is 5, because 5 × 5 = 25. The sign is √.
Powers and indices
A power (also called an index or exponent) is a small raised number showing how many times to multiply a number by itself. So 2⁴ means 2 × 2 × 2 × 2 = 16.
Times tables
Times tables are lists of the multiplication facts for a number, such as 3 × 1 = 3 up to 3 × 12 = 36. Knowing them by heart speeds up almost all later maths.
Area
Area is the amount of flat space inside a shape, measured in square units like cm² or m². A rectangle 4 cm by 3 cm has an area of 4 × 3 = 12 cm².
Prime numbers
A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13.
Cube numbers and cube roots
A cube number is a number multiplied by itself three times, like 4³ = 4 × 4 × 4 = 64. The cube root undoes it, so ∛64 = 4.