√ Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. It's a fundamental concept in mathematics, allowing you to solve equations, calculate geometric dimensions, and understand real-world measurements.
The square root (√) is one of the most essential mathematical operations used by students, teachers, engineers, and scientists worldwide. It serves as the main tool for finding side lengths in geometry, calculating standard deviations in statistics, and solving quadratic equations.
Simply enter any number into our square root calculator above to instantly find its root, simplify radicals, or convert it to a decimal. Understanding how it works helps you build a stronger foundation in math and apply it confidently in real life.
Top Square Root Calculations
Click any number to instantly calculate its square root. These are the most searched square roots on the web.
What Is a Square Root and Why It Matters
If you have ever needed to find the length of a side of a square, calculate the distance between two points, or solve a complex equation, you've likely used the square root function.
This mathematical operation acts like a reverse multiplier — finding the base number that was multiplied by itself. It's commonly used in various fields such as architecture, physics, computer graphics, and finance to make sense of squared numbers.
In simple words, it asks the question: "What number, when multiplied by itself, gives me this number?" It lets you reverse the process of squaring a number — taking you from the area of a shape back to its side length.
Common Square Roots You Should Know
Different numbers have different types of square roots. Some are perfect squares (whole numbers), while others are irrational numbers (infinite decimals).
| Number | Square Root (√) | Decimal Form |
|---|---|---|
| 100 | 10 | 10.0 |
| 225 | 15 | 15.0 |
| 64 | 8 | 8.0 |
| 50 | 5√2 | 7.071... |
| 36 | 6 | 6.0 |
| 32 | 4√2 | 5.656... |
| 20 | 2√5 | 4.472... |
| 16 | 4 | 4.0 |
| 12 | 2√3 | 3.464... |
| 10 | √10 | 3.162... |
| 8 | 2√2 | 2.828... |
| 7 | √7 | 2.645... |
| 6 | √6 | 2.449... |
| 5 | √5 | 2.236... |
| 3 | √3 | 1.732... |
| 2 | √2 | 1.414... |
How to Calculate the Square Root
Finding the square root is easier than you might think. You can use a calculator for instant answers, or learn manual methods to understand the math behind it.
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1
Open our square root calculator tool at the top of this page.
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Type your number into the input field and press
Calculate. -
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The exact radical form and decimal approximation will appear instantly.
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For manual calculation, find the largest perfect square less than your number, factor it out, and simplify the remaining radical.
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Use the results for geometry, algebra equations, physics problems, and much more.
What You Can Do with Square Roots
Once you understand how square roots work, you'll find a wide range of applications. These tools help you solve practical problems across multiple disciplines.
Geometry & Area
Find the side length of a square or rectangle if you only know the area. If a square has an area of 64 cm², each side is √64 = 8 cm.
Distance Formula
Calculate the exact distance between two points on a graph using the Pythagorean theorem: d = √((x₂-x₁)² + (y₂-y₁)²).
Quadratic Equations
Solve for unknown variables using the quadratic formula, which relies on the square root to find two possible solutions.
Square Root Curves
Adjust test scores or data sets using a square root curve to fairly boost lower scores while preserving rank order.
Standard Deviation
Analyze data spread in statistics by taking the square root of the variance — a critical step in data science.
Physics & Engineering
Determine velocity, voltage, and force in various scientific formulas where squared values need to be reversed.
Common Problems with (and Easy Fixes)
Even when the concept seems simple, small errors can occur while calculating or writing square roots.
The Decimal Goes On Forever
If you get a decimal that never ends (like 1.414... for √2), it's an irrational number. Leave it in radical form (e.g., √2) for exact math, or round to 2–3 decimal places for practical use.
Confusing Squaring with Square Rooting
A common mistake is thinking the square root of 9 is 81 (that's squaring). Remember: Square root reduces the number (√9 = 3), squaring multiplies it (9² = 81).
Negative Numbers Under the Root
You can't take the square root of a negative number in basic math (it results in an imaginary number). Check your equations for negative values before calculating.
Forgetting to Simplify Radicals
Writing √8 instead of 2√2 is technically correct but not fully simplified. Always pull out perfect square factors (like 4 from 8) to make your answers cleaner.
Calculating the Square Root Curve Wrong
When applying a square root curve to a grade, make sure you multiply the final result by 10. Formula: Grade = 10 × √(Original Score). For example, √81 = 9, then 9 × 10 = 90%. For a score of 36: √36 = 6, then 6 × 10 = 60%.
How to Type the Square Root Symbol on Any Device
If you need to write the √ symbol but don't know how, here's how to easily type it on any platform.
Windows
- Open your document and hold down
Alt - Type
251on the numeric keypad - Release the key — the √ symbol appears
macOS
- Press
Control + Cmd + Space - Search for "square root"
- Double-click the √ symbol to insert it
Android
- Tap
?123to switch to symbols - Tap the
=\<key for more symbols - Select the √ symbol from the extended keyboard
iPhone (iOS)
- Go to Settings → General → Keyboard → Text Replacement
- Add shortcut: type "sqrt" as the phrase
- Now typing "sqrt" auto-replaces with √
Why the Square Root of a Negative Number is "Imaginary"
It's a common point of confusion, but basic square roots only apply to positive numbers. If you try to find the square root of -9, no real number multiplied by itself equals a negative.
| Invalid Concept | Correct Concept |
|---|---|
| √-16 = 4 | √-16 = 4i |
| √-1 = -1 | √-1 = i |
| √(-25) = 5 | √(-25) = 5i |
Square Root Curves in Grading
A square root curve is a popular method teachers use to adjust test scores fairly. It gives a larger boost to students who scored lower, while giving minimal boost to high-scoring students.
- Boost low scores significantly (e.g., a 36 becomes a 60)
- Reward high scores slightly (e.g., a 100 stays a 100)
- Maintain the rank order of the class
- Apply a mathematically justified grading adjustment
- Formula: Final Grade = 10 × √(Original Score)
Tips for Simplifying Square Roots Fast
Simplifying radicals doesn't have to be complicated — it just requires knowing your perfect squares.
Memorize Perfect Squares
Know 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 by heart. This is the single most useful thing you can do.
Factor the Radicand
Break the number down into a perfect square multiplied by another number. For √72, think 72 = 36 × 2, so √72 = 6√2.
Pull Out the Pair
Move the square root of the perfect square outside the radical. √50 = √(25 × 2) = 5√2. Always simplify fully.
Combine Like Terms
When adding or subtracting, only combine radicals with the same number inside: 3√2 + 5√2 = 8√2. You can't combine √2 + √3.
Rationalize the Denominator
Never leave a square root in the bottom of a fraction. Multiply top and bottom by the radical to remove it from the denominator.
Estimate Quickly
Find the two closest perfect squares. For √70, since 64 < 70 < 81, the answer is between 8 and 9 — closer to 8.4.
Frequently Asked Questions
Final Thoughts
The square root is your key to unlocking a deeper understanding of mathematics. It allows you to reverse squared numbers, solve complex geometric problems, and apply logic to real-world scenarios — all from one simple symbol.
- The square root of x is the number that equals x when multiplied by itself.
- Always simplify radicals by pulling out perfect squares.
- Keep the √ symbol handy for clean, exact mathematical communication.
Think of the square root as your mathematical reverse gear — when used wisely, it makes solving equations, analyzing data, and understanding the world around you much simpler and more precise.