Square Root Table
This square root table provides exact and approximate values for numbers from 1 to 100. It is a quick reference for students, engineers, scientists, and anyone working with geometry, physics, or statistics who needs square roots without using a calculator.
What Is a Square Root?
A square root of a number n is a value that, when multiplied by itself, equals n.
For example:
√9 = 3because3 × 3 = 9√2 ≈ 1.4142√10 ≈ 3.1623
Mathematically:
√n = x if x2 = n.
Square roots appear in geometry, algebra, physics, statistics, and many other fields.
Square Root Table (Selected Values from 1–100)
The following table shows approximate square roots for some common values between 1 and 100.
| Number (n) | √n (Approx.) | Perfect Square? |
|---|---|---|
| 1 | 1.0000 | Yes |
| 2 | 1.4142 | No |
| 3 | 1.7320 | No |
| 4 | 2.0000 | Yes |
| 5 | 2.2360 | No |
| 6 | 2.4490 | No |
| 7 | 2.6458 | No |
| 8 | 2.8284 | No |
| 9 | 3.0000 | Yes |
| 10 | 3.1623 | No |
| 11 | 3.3166 | No |
| 12 | 3.4641 | No |
| 13 | 3.6056 | No |
| 14 | 3.7417 | No |
| 15 | 3.8730 | No |
| 16 | 4.0000 | Yes |
| 17 | 4.1231 | No |
| 18 | 4.2426 | No |
| 19 | 4.3589 | No |
| 20 | 4.4721 | No |
| 25 | 5.0000 | Yes |
| 36 | 6.0000 | Yes |
| 49 | 7.0000 | Yes |
| 64 | 8.0000 | Yes |
| 81 | 9.0000 | Yes |
| 100 | 10.0000 | Yes |
A full printable version of the square root table (1 to 100) is available: download the square root reference table (PDF) .
Perfect Squares Table (1² to 31²)
Perfect squares are numbers that can be written as the square of an integer. They are useful for quickly estimating or bounding square roots.
| n | n² |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
| 16 | 256 |
| 17 | 289 |
| 18 | 324 |
| 19 | 361 |
| 20 | 400 |
| 21 | 441 |
| 22 | 484 |
| 23 | 529 |
| 24 | 576 |
| 25 | 625 |
| 26 | 676 |
| 27 | 729 |
| 28 | 784 |
| 29 | 841 |
| 30 | 900 |
| 31 | 961 |
How to Estimate a Square Root (Without a Calculator)
To estimate √n manually, you can follow these steps:
- Identify the two closest perfect squares. For example, 50 lies between 49 and 64, so
√50is between 7 and 8. - Use a simple approximation formula. One common method is:
√n ≈ a + (n - a²)/(2a), whereais the square root of the lower perfect square.
Example for √50:
Here, a = 7, because 7² = 49 and 49 < 50 < 64.
√50 ≈ 7 + (50 - 49) / (2 × 7) = 7 + 1 / 14 ≈ 7.071
This gives a surprisingly accurate approximation to the true value.
Common Square Roots to Remember
The following square roots appear very often in math, physics, and engineering problems.
- √2 ≈ 1.4142
- √3 ≈ 1.7320
- √5 ≈ 2.2360
- √10 ≈ 3.1623
- √50 ≈ 7.0711
- √100 = 10
Applications of Square Roots
Square roots appear in many real-world calculations and formulas across different fields.
Geometry
- Distance formula in the coordinate plane
- Pythagorean theorem
- Scaling shapes and areas
Physics
- Energy and power formulas
- Equations of motion
- Wave and oscillation analysis
Statistics
- Variance and standard deviation
- Regression and error analysis
Engineering
- Signal processing
- Load and stress calculations
- Scaling models and simulations
Frequently Asked Questions (FAQ)
What is a square root in simple terms?
A square root is a number that multiplies by itself to give another number. For example, 4 is a square root of 16 because 4 × 4 = 16.
What are perfect squares?
Perfect squares are numbers such as 1, 4, 9, 16, 25, and so on, that can be written as the square of an integer (1², 2², 3², etc.).
Why do some numbers not have an exact square root?
Many numbers have square roots that are irrational, meaning their decimal expansion is non-terminating and non-repeating. For example, √2 cannot be written as a simple fraction.
What is the square root of 2?
The square root of 2 is approximately 1.4142. It is an irrational number and appears in many geometric formulas.
What are square roots used for?
Square roots are used in geometry, physics, engineering, statistics, finance, and many other areas, especially when working with distances, areas, variances, and scaling.