Key takeaways
- √n is the value that, multiplied by itself, gives n.
- Check a perfect square first: 1, 4, 9, 16, 25, 36... have whole-number square roots.
- Negative numbers have no real square root.
- Squaring your answer should give back the original number.
In this guide
What is a square root?
The square root of a number n is the value that, multiplied by itself, gives n. For example, the square root of 9 is 3, because 3 × 3 = 9. The square root symbol is √.
Why does the square root matter?
Square roots undo squaring, which shows up constantly in area, distance, and growth problems. Whenever you know an area and need a side length, or a squared distance and need the real distance, you need a square root.
The square root of 16 is 4, not 8. Squaring 4 (4 × 4 = 16) proves it — dividing 16 by 2 is a completely different operation.
The formula
| Symbol | What it means | Example |
|---|---|---|
| √ | The square root symbol | — |
| n | The number you're finding the square root of (must be 0 or positive) | 225 |
How to find a square root step by step
- 1Check if the number is a perfect square, meaning a whole number times itself equals it (like 1, 4, 9, 16, 25, 36...).
- 2If it is, find the whole number that, multiplied by itself, gives your number.
- 3If it isn't a perfect square, estimate between the two nearest perfect squares, or use a calculator for a decimal answer.
- 4Check your answer by squaring it (multiplying it by itself) — it should equal the original number.
Enter your own number in the calculator below to see the check step laid out.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Trying to find the square root of a negative number. There's no real number that squares to a negative result, so this is undefined in ordinary math.
- Forgetting units. If n is an area, like 225 m², the square root gives a length, like 15 m, not a squared unit.
- Guessing without checking. Always square your estimate to confirm it matches the original number.
Tips and tricks
- Estimate a non-perfect square by finding the two nearest perfect squares and judging which one it's closer to.
- Double-check any decimal answer by squaring it on a calculator — it should land close to the original number.
- Square roots power the Pythagorean theorem and solving a quadratic equation.
Where you'll use it in real life
- Home and garden: finding the side length of a square plot from its area.
- Construction: calculating distances using the Pythagorean theorem.
- Math class: solving quadratic equations, which always involve a square root step.
- Design: working out a square screen or tile size from its total area.
Quick summary
To find a square root, check whether the number is a perfect square, then find (or estimate) the value that, squared, gives it back. Negative numbers have no real square root. For a cube instead of a square, see the cube root calculator. Try the calculator above with your own number.
Worked example
A square community garden has an area of 225 square meters. How long is each side of the garden?
- 01√225
- 0215 × 15 = 225
- 03√225 = 15
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I find the square root of a number that isn't a perfect square?
Find the two nearest perfect squares to estimate a range, then use a calculator for a precise decimal value. For example, √20 is between √16 (4) and √25 (5), closer to 4.47.
Can a negative number have a square root?
Not a real one. Since a negative times a negative is positive, no real number squares to give a negative result, so square roots of negative numbers are undefined in real numbers.
What is a perfect square?
A perfect square is a number you get by multiplying a whole number by itself, like 1, 4, 9, 16, and 25. Their square roots are whole numbers.
What is the square root of 0?
It's 0, because 0 × 0 = 0. Zero is the only number whose square root equals itself besides 1.
