Square Root Calculator Guide | How to Calculate √

What is a Square Root?

A square root is the inverse operation of squaring a number. When you square a number, you multiply it by itself. The square root asks: "What number, when multiplied by itself, gives this result?"

For example, the square root of 16 is 4, because 4 × 4 = 16. We write this as √16 = 4.

Perfect Squares (1-225)

Perfect squares are numbers that have exact whole number square roots. These are worth memorising:

Extended Perfect Squares

Methods to Calculate Square Roots

Method 1: Estimation Between Perfect Squares

Method 2: Babylonian Method (Newton's Method)

This iterative method quickly converges to the correct answer:

Method 3: Prime Factorisation

Useful for simplifying square roots and finding exact roots of some numbers:

Common Square Roots

Properties of Square Roots

Examples Using Properties

• √(4 × 9) = √4 × √9 = 2 × 3 = 6
• √(100/25) = √100 / √25 = 10 / 5 = 2
• √(16 × 5) = √16 × √5 = 4√5 ≈ 8.944

Practical Applications

1. Calculating Distances (Pythagorean Theorem)

2. Finding the Side of a Square

3. Standard Deviation in Statistics

The standard deviation involves taking the square root of the variance.

4. Physics and Engineering

• RMS (Root Mean Square) values for AC electricity
• Speed calculations from kinetic energy
• Wave equations

Cube Roots and Higher

The same concept extends to cube roots and nth roots:

Perfect Cubes

GCSE and A-Level Topics

Simplifying Surds

A surd is an irrational root that cannot be simplified to a whole number:

Rationalising the Denominator

Frequently Asked Questions

What is a square root?

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 × 5 = 25. The symbol for square root is √.

How do you calculate a square root by hand?

One method is estimation and refinement: find the perfect squares on either side of your number, then estimate between them. For example, √50 is between √49 (7) and √64 (8), closer to 7. A better estimate is 7.07. You can refine using the formula: new estimate = (estimate + number/estimate) ÷ 2.

What are perfect squares?

Perfect squares are numbers that have whole number square roots. The first 15 perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. These are useful to memorise for quick mental calculations.

Can you have a square root of a negative number?

Not in real numbers. The square root of a negative number is an imaginary number. For example, √-1 is defined as 'i' (the imaginary unit). √-9 = 3i. These are used in complex numbers and advanced mathematics.

What is √2?

√2 (approximately 1.41421) is an irrational number – it cannot be expressed as a simple fraction and its decimal goes on forever without repeating. It's the length of the diagonal of a unit square.

Square Roots in UK Education and Everyday Applications

Square roots are formally introduced in the UK national curriculum during Key Stage 3 (ages 11 to 14) and are an essential topic for GCSE Mathematics, which is taken by virtually all students in England, Wales, and Northern Ireland at age 15 or 16. The Edexcel, AQA, and OCR exam boards all include square root questions in both foundation and higher tier papers. At foundation level, students are expected to find square roots of perfect squares and estimate square roots of non-perfect squares. At higher level, questions extend to simplifying surds (expressions involving square roots) and rationalising denominators, topics that also appear in A-level Mathematics.

Beyond the classroom, square roots have practical applications that UK residents encounter regularly. Calculating the diagonal measurement of a room or screen uses the Pythagorean theorem, which requires a square root. Television screens sold in the UK are measured diagonally in inches; knowing the width and height allows you to calculate the actual diagonal using the formula: diagonal equals the square root of (width squared plus height squared). Similarly, when laying flooring or fitting carpets, tradespeople use square roots to determine diagonal measurements for cutting and fitting. Builders and architects use square root calculations extensively when working with the right-angle triangle relationships that underpin structural engineering and design.

In UK financial contexts, square roots appear in compound interest and investment calculations. The geometric mean return of an investment, which gives a more accurate picture of average annual growth than the arithmetic mean, requires taking the nth root (a generalisation of the square root). UK financial advisers use these calculations when assessing pension fund performance, ISA returns, and portfolio growth over multiple years. The formula for annualised return is: annualised return equals the nth root of (final value divided by initial value) minus 1, where n is the number of years. Understanding square roots helps UK investors evaluate whether their investments are genuinely outperforming inflation and meeting their retirement savings goals.

Practical Tips for Working with Square Roots

• Memorise common perfect squares: Knowing that 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144 are perfect squares (1 through 12 squared) makes estimation and mental calculation much faster.
• Estimate between known values: To estimate the square root of 50, note that 7 squared is 49 and 8 squared is 64. Since 50 is very close to 49, the square root of 50 is approximately 7.07.
• Use square roots for area-to-side conversions: If you know the area of a square space in square metres or square feet, taking the square root gives the length of each side, which is useful for measuring rooms and garden plots.

More Questions About Square Roots