# 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

| n | \( \sqrt{n} \) (exact) | \( \sqrt{n} \) (4 d.p.) |
| --- | --- | --- |
| 1 | 1 | 1.0000 |
| 2 | \( \sqrt{2} \) | 1.4142 |
| **3** | \( \sqrt{3} \) | 1.7321 |
| 4 | 2 | 2.0000 |
| 5 | \( \sqrt{5} \) | 2.2361 |
| 6 | \( \sqrt{6} \) | 2.4495 |
| 7 | \( \sqrt{7} \) | 2.6458 |
| 8 | \( \sqrt{8} \) | 2.8284 |
| 9 | 3 | 3.0000 |
| 10 | \( \sqrt{10} \) | 3.1623 |
| 11 | \( \sqrt{11} \) | 3.3166 |
| 12 | \( \sqrt{12} \) | 3.4641 |
| 13 | \( \sqrt{13} \) | 3.6056 |
| 14 | \( \sqrt{14} \) | 3.7420 |
| 15 | \( \sqrt{15} \) | 3.8729 |
| 16 | 4 | 4.0000 |
| 17 | \( \sqrt{17} \) | 4.1231 |
| 18 | \( \sqrt{18} \) | 4.2426 |
| 19 | \( \sqrt{19} \) | 4.3589 |
| 20 | \( \sqrt{20} \) | 4.4721 |

\( \sqrt{3} \) 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: \( a + b \neq \sqrt{a} + \sqrt{b} \).