How to Find the Value of Root 3 and Its Uses in Maths
The concept of Value of Root 3 (√3) plays a key role in mathematics and is widely applicable to geometry, trigonometry, and various exam situations. Accurate knowledge of √3 helps students quickly solve problems involving triangles, area, and trigonometric values.
What Is Value of Root 3 (√3)?
The Value of Root 3 (also written as √3) is a mathematical constant. It is defined as the positive number that, when multiplied by itself, equals 3. In mathematical terms, √3 × √3 = 3. You’ll find this concept applied in trigonometry (especially in sin 60°, tan 60°), geometry (like the area of equilateral triangles), and measurement problems involving diagonals or altitudes.
Decimal Value of Root 3
| Root | Decimal Value | Surd/Exact Form |
|---|---|---|
| √2 | 1.414 | √2 |
| √3 | 1.732 | √3 |
| √5 | 2.236 | √5 |
The value of root 3 up to three decimal places is 1.732. It is an irrational number, so its actual decimal expansion continues forever without repeating. For most school exams and calculations, you can safely use 1.732 or 1.73 as the value of √3 unless more precision is required.
How to Calculate Square Root of 3?
Root 3 is not a perfect square, so we use the long division method to estimate its value. Here are the steps:
- Write 3 as 3.000000, grouping zeros in pairs after the decimal.
- Find the biggest integer whose square is ≤ 3 (that’s 1). Place 1 in the quotient; 1 × 1 = 1. Subtract: 3 – 1 = 2.
- Bring down the first pair of zeros. The new dividend is 200.
- Double the quotient: 2 × 1 = 2. Write it as the new divisor’s first digit (20_).
- Find a digit to fit as the next quotient and divisor digit so (20_ × _) is ≤ 200 (answer: 27).
- Continue with more pairs of zeros for more decimal places.
After two steps, you get √3 ≈ 1.73. More steps will give √3 ≈ 1.73205.
Why Is Root 3 (√3) Irrational?
Root 3 cannot be expressed as a fraction of two whole numbers. Its decimal part does not terminate or repeat, making it an irrational number. In proofs, we show that if √3 were rational, then 3 would have to divide the square of any numerator in its fraction form, leading to a contradiction unless it was a perfect square, which it is not.
Where Is Value of Root 3 Used?
- Trigonometric ratios: sin 60° = √3/2, tan 60° = √3
- Area of equilateral triangle: (√3 / 4) × side²
- Altitude/height of equilateral triangle: (√3 / 2) × side
- Diagonals in cube/cuboid or finding lengths in geometry
- Formulas in physics and electrical engineering (like 3-phase power)
Key Formula Block Involving Root 3 (√3)
- Area of equilateral triangle = (√3/4) × a²
- Height = (√3/2) × a
- sin 60° = √3 / 2
- tan 60° = √3
Solved Example Using Root 3
Example 1: Find the height of an equilateral triangle whose side is 6 cm.
2. Substitute side: Height = (√3/2) × 6 = 3√3 cm
3. Substitute √3 ≈ 1.732: Height ≈ 3 × 1.732 = 5.196 cm
4. Final Answer: Height ≈ 5.196 cm
Example 2: What is the value of tan 60°?
2. Use decimal: tan 60° ≈ 1.732
Tips & Tricks to Remember Value of Root 3
- Remember: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236
- A triangle with sides 1-√3-2 is often used for sin and tan 60°
- Write key roots on a formula page for last-minute revision
- For MCQs, use 1.73 or 1.732; for higher accuracy use more decimals
Common Pitfalls and Misunderstandings
- Forgetting that √3 is irrational, not a simple fraction
- Using too few decimal places when more accuracy is needed
- Mixing up root 3 (√3) with root 2 (√2), especially in geometry/trig
How Value of Root 3 Links to Other Concepts
Knowing the value of root 3 is essential for solving area and height problems in triangles, especially equilateral and 30-60-90 ones. It’s also central in learning trigonometric values and comparing with root 2 and root 5. For more on calculation methods, see square root long division method.
Quick Classroom Tip
Link the number 1.732 to the 60° angle in triangles—whenever you see tan 60° or sin 60°, recall that √3 appears in their values. Vedantu teachers use memory tricks and tables to help students remember such roots easily during sessions.
We explored the value of root 3—from definition, decimal expansion, calculation methods, main uses, and solved examples, to common mistakes and tricks to remember it. Keep practicing and use live classes or revision notes from Vedantu to master root values for exams and beyond.
Recommended Reads: Value of Root 2 | Value of Root 5 | Square Root Long Division Method | Trigonometric Ratios of Standard Angles
FAQs on Value of Root 3 (√3): Decimal, Formula & Applications
1. What is the value of root 3 (√3)?
The value of root 3 (√3) is approximately 1.732. It's an irrational number, meaning its decimal representation is non-terminating and non-repeating. This value is frequently used in various mathematical calculations, particularly in geometry and trigonometry.
2. Is root 3 a rational or irrational number?
Root 3 (√3) is an irrational number. This means it cannot be expressed as a simple fraction of two integers (a/b, where a and b are integers and b ≠ 0).
3. How do you calculate the square root of 3?
You can calculate the square root of 3 using several methods:
• The long division method provides a stepwise approach to finding increasingly accurate decimal approximations.
• Using a scientific calculator is the most straightforward method for obtaining a precise value.
• Approximation methods can be used to quickly estimate the value. For instance, knowing that 1.7² = 2.89 and 1.8² = 3.24 helps narrow it down quickly.
4. What is the decimal value of root 3?
The decimal value of root 3 (√3) is approximately 1.7320508... The decimal representation continues infinitely without repeating.
5. How is root 3 used in trigonometry?
Root 3 appears in the trigonometric ratios of certain standard angles. For example:
• sin 60° = √3/2
• cos 30° = √3/2
• tan 60° = √3
6. Where is root 3 used in geometry?
Root 3 is commonly found in formulas related to equilateral triangles. For example, the altitude of an equilateral triangle with side length 'a' is given by (√3/2)a, and its area is (√3/4)a². It also appears in calculations involving other special triangles.
7. What are some formulas involving √3?
Some important formulas involving √3 include:
• Area of an equilateral triangle: A = (√3/4)a² (where 'a' is the side length)
• Altitude of an equilateral triangle: h = (√3/2)a (where 'a' is the side length)
• Trigonometric ratios: sin 60° = cos 30° = √3/2 and tan 60° = √3
8. How accurate does the value of root 3 need to be for typical calculations?
The required accuracy for √3 depends on the context. For most school-level problems, using 1.732 is sufficient. However, for more advanced calculations or engineering applications, you might need more decimal places for greater precision.
9. What is the relationship between root 3 and other common root values (like √2)?
While not a direct mathematical relationship, √2, √3, and √5 are all frequently encountered irrational numbers often studied together in mathematics due to their appearance in various geometric problems and trigonometric identities.
10. Are there any tricks or mnemonics to remember the value of root 3?
There are various memorization techniques, such as using flashcards, creating rhymes, or connecting the value to easily remembered numbers and patterns. Repeating the value and using it in practice problems is the best way to solidify your understanding. Remember that using approximations and techniques is an important aspect of mathematics.
11. Can the value of root 3 be expressed as a simple fraction?
No, because √3 is an irrational number; it cannot be expressed as a fraction of two integers.
