Square Root of 3 — Value, How to Find It, and Examples

TL;DR

The square root of 3 is about 1.732, and because 3 is prime, ( \sqrt{3} ) is irrational and already in simplest radical form. This article gives the exact and decimal value, two ways to compute it by hand, where ( \sqrt{3} ) turns up in geometry, and the mistakes students make most.

The Answer at a Glance

Quick Answer:

Result: ( \sqrt{3} \approx 1.7320508 )

Notation: Decimal approximation; exact form is ( \sqrt{3} ).

Method shown: Long division by hand, cross-checked with estimation between perfect squares.

Approximate value: 1.7321 (4 d.p.)

Exact form: ( \sqrt{3} ) — cannot be simplified, since 3 is prime.

Quick Reference Table — Square Roots From 1 to 20

( \sqrt{3} ) sits between 1 and 2, and a little past the midpoint because 3 is closer to 4 than to 1.

Where √3 Appears

( \sqrt{3} ) is the height of an equilateral triangle whose side length is 2 — drop a perpendicular and Pythagoras gives ( \sqrt{2^2 - 1^2} = \sqrt{3} ). It is also the length of the space diagonal of a unit cube, and shows up constantly in trigonometry too: ( \tan 60^{\circ} = \sqrt{3} ).

What "square root of 3" Means

The square root of a non-negative number ( n ) is the value ( x ) such that ( x^2 = n ). For ( \sqrt{3} ), it is the positive ( x ) with ( x^2 = 3 ).

Because ( 1^2 = 1 ) and ( 2^2 = 4 ), the answer must land between 1 and 2 — and squaring 1.732 gives approximately 3, which confirms the value.

Is The Square Root of 3 Rational or Irrational?

( \sqrt{3} ) is irrational. A whole number is a perfect square only when every prime in its factorization appears an even number of times; 3 is prime, so it carries the single factor 3 — an odd exponent — and cannot be a perfect square.

The classic proof by contradiction makes this airtight: assume ( \sqrt{3} = \frac{p}{q} ) in lowest terms, square to get ( 3q^2 = p^2 ), and the parity of the factor 3 on each side cannot match. The decimal 1.7320508… never terminates and never repeats.

How To Find √3 — Two Methods

Method 1 — Long division (digit by digit)

Write 3 as 3.000000 and pair the digits after the decimal point.

Step 1. The largest integer whose square is at most 3 is 1. Subtract: 3−1=2. Bring down 000000 to get 200.

Step 2. Double the quotient 1 to get 2. Find ( d ) with ( (20+d) \cdot d \leq 200 ). Here ( d=7 ) gives ( 27 \cdot 7 = 189 ). Subtract: 200−189=11. Bring down 000000 to get 1100.

Step 3. Continue this process to refine the approximation.

Final answer: ( \sqrt{3} \approx 1.7321.

Method 2 — Estimation between perfect squares

Start at 1.7: ( 1.72 = 2.89 ) is a touch low. Try ( 1.73: ) ( 1.732 = 2.9929 ). Each refinement nudges the guess upward toward 3, landing on 1.732 for everyday use.

What Are The Most Common Mistakes With √3?

Mistake 1: Treating √3 as if it can be simplified

Don't do this: Writing ( \sqrt{3} = 1 \cdot \sqrt{3} ) and then "simplifying" further.

The correct way: ( \sqrt{3} ) is already in simplest radical form.

Mistake 2: Splitting the root over addition

Don't do this: ( \sqrt{1 + 2} = \sqrt{1} + \sqrt{2} ).

The correct way: Square roots do not distribute over addition.

Mistake 3: Rounding too early

Don't do this: Replacing ( \sqrt{3} ) with an approximate value too early in calculations.

The correct way: Carry the exact form through the algebra and convert to a decimal only at the final step.

Examples of Square Root of 3

Example 1

Simplify ( \sqrt{12} ) using ( \sqrt{3} ).

( \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \approx 3.4641.

Example 2 (Wrong path first)

Find the height of an equilateral triangle with side 2.

Wrong attempt: Student writes the height as half the side.

Correct: Drop the altitude to split the base in half: height = ( \sqrt{2^2 - 1^2} = \sqrt{3} \approx 1.732.

Example 3

Evaluate ( \tan 60^{\circ} ).

In a 30-60-90 triangle, the sides are in ratio 1:( \sqrt{3} ):2, so ( \tan 60^{\circ} = \sqrt{3} \approx 1.732.

Example 4

Rationalise ( \frac{1}{\sqrt{3}} ).

Multiply top and bottom by ( \sqrt{3} ): ( \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \approx 0.5774.

Example 5

Find the space diagonal of a cube with side 1.

The diagonal is ( \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \approx 1.732.

Conclusion

• The square root of 3 is approximately 1.732 — irrational, non-terminating, non-repeating.
• 3 is prime, so ( \sqrt{3} ) cannot be simplified and stays in radical form.
• Long division and estimation both pin the value down by hand.
• Square roots do not distribute over addition: ( a + b \neq \sqrt{a} + \sqrt{b} ).