Square root of 3
√3 is about 1.732. It is not a whole number, and here is why.
The answer
The square root of 3 is about 1.732. It is between 1 and 2, because 1 < 3 < 4.
3 drawn as a square
3 counters will not make a square. 1 of them makes a 1 by 1 square, and 2 are left over. A 2 by 2 square needs 1 more. So the side, √3, is more than 1 but less than 2.
Between 1 and 2
1 < 3 < 4, so 1 < √3 < 2
1 × 1 = 1, which is too small, and 2 × 2 = 4, which is too big. So √3 is between 1 and 2. 3 is 2 away from 1 and 1 away from 4, so √3 is nearer 2. It is about 1.732, or 1.73 to 2 decimal places.
Why the decimal never ends
Check the answer by multiplying it by itself: 1.732 × 1.732 = 2.999824. That is very close to 3, but not quite. Adding more decimal places gets closer and closer, but never exactly there.
√3 is an irrational number: its digits go on forever without a repeating pattern, so no fraction is exactly equal to it. That is why we write √3 for the exact value and 1.732 when we need a number to work with.
Where √3 turns up
- An equilateral triangle with sides 2 cm long is exactly √3 cm tall, about 1.73 cm.
- The longest diagonal of a cube with edges 1 cm long, from one corner right through to the opposite corner, is √3 cm.
Square numbers and their roots
The square numbers up to 144 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Children in England learn to recognise them in Year 5, and use them all through the 11+.
More numbers
See every factor of 3 or check whether 3 is prime.
Questions parents ask
What is √3 to 1 decimal place, and to 5?
1.7 to 1 decimal place, and 1.73205 to 5 decimal places.
A square has an area of 3 cm². How long is each side?
About 1.73 cm. The exact length is √3 cm, which no ruler can measure exactly, because its decimal never ends.
Can a square root be negative?
Yes. −√3 × −√3 also makes 3, so −√3 is a square root of 3 too. At primary school, "the square root" means the positive one, √3. Multiplying negative numbers comes later, at secondary school.
Number facts are the base of 11+ maths
Factors, multiples, primes and square numbers come up all over 11+ maths, from simplifying fractions to ratio. The free diagnostic checks maths, English and both kinds of reasoning, and shows what to practise next. No card is required.