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.
A note about this topic
England’s Year 5 curriculum includes recognising square numbers. The square-root explanations on this page are extension material.
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.
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 mathematical length is √3 cm. Use an approximation for a measurement.
Can a square root be negative?
Yes. −√3 × −√3 also makes 3, so −√3 is a square root of 3 too. For a positive number, the symbol √ denotes the positive root, √3. The negative root also gives 3 when multiplied by itself.
Continue your number practice
Explore the diagnostic for another set of practice questions.