Square Root 3 Calculator App

What is the square root of 3?

The square root of 3 is the number which, when multiplied by itself, gives exactly 3. Mathematically, this is represented as:

Since 3 is not a perfect square, √3 cannot be expressed as a simple fraction. It's an irrational number that continues infinitely without repeating. The decimal approximation of √3 is approximately 1.73205080757.

This value appears frequently in geometry, trigonometry, and engineering calculations, particularly in problems involving equilateral triangles and 30-60-90 right triangles.

How to calculate √3

There are several methods to calculate the square root of 3:

1. Using a calculator

The simplest method is to use a scientific calculator or this online calculator app. Simply input 3 and press the square root button to get the result.

2. Using the long division method

For a more manual approach, you can use the long division method:

1. Write 3 as 3.000000
2. Find the largest number whose square is less than 3 (which is 1)
3. Subtract 1 from 3 to get 2
4. Bring down two zeros to make it 200
5. Double the current result (1) to get 2, and find a digit to place after it so that (20 × digit) is less than or equal to 200
6. 7 works because 20 × 7 = 140, which is less than 200
7. Subtract 140 from 200 to get 60
8. Bring down two more zeros to make it 6000
9. Double the current result (17) to get 34, and find a digit so that (340 × digit) is less than or equal to 6000
10. 8 works because 340 × 8 = 2720, which is less than 6000
11. Continue this process to get more decimal places

This method will give you √3 ≈ 1.73205080757.

3. Using a series expansion

Mathematically, √3 can be expressed using the binomial series expansion:

This infinite series converges to √3 when summed to infinity.

Practical applications

The square root of 3 has several practical applications in various fields:

1. Geometry

In an equilateral triangle with side length 2, the height is √3. This is because:

2. Trigonometry

The value of √3 appears in trigonometric functions of 30°, 60°, and 90° angles. For example:

3. Engineering

In electrical engineering, √3 is used in calculations involving three-phase systems. The line voltage in a balanced three-phase system is √3 times the phase voltage.

4. Physics

√3 appears in calculations involving wave phenomena and quantum mechanics. For example, in the double-slit experiment, the interference pattern depends on √3 for certain configurations.

Square root of 3 in math

In mathematics, √3 is a fundamental constant with several important properties:

• It's an irrational number, meaning it cannot be expressed as a simple fraction
• It's transcendental, meaning it's not the root of any non-zero polynomial equation with rational coefficients
• It's approximately 1.73205080757
• It's the side length of an equilateral triangle with unit height

√3 is also related to the golden ratio φ (phi) through the equation:

Notice that √3 is slightly larger than φ, which has implications in geometry and art.

Square root of 3 in engineering

In engineering, √3 is particularly important in electrical engineering and structural design:

1. Three-phase power systems

In three-phase AC power systems, the relationship between line voltage (VL) and phase voltage (VP) is given by:

This is because the three-phase voltages are 120° apart, creating a vector sum that's √3 times larger than any single phase voltage.

2. Structural engineering

In truss bridges and other structures, √3 appears in calculations involving equilateral triangles and 30-60-90 triangles. For example, the height of a truss member in an equilateral triangle configuration is √3 times the side length.

3. Antenna design

In antenna theory, √3 is used in calculations involving the spacing of elements in phased arrays. The optimal spacing between elements often involves √3 for certain antenna patterns.