Math.SQRT1_2

Added in vES1 · Updated March 15, 2026 · Math

javascript math constants square-root sqrt1-2

Math.SQRT1_2 is a static property of the Math object that represents the square root of 1/2, approximately 0.7071067811865476. It is a read-only constant; you cannot change its value.

Mathematically, this equals 1 / Math.sqrt(2) or Math.sqrt(2) / 2. It appears frequently in trigonometry (cos 45° = sin 45°), normalization calculations, and probability distributions.

Syntax

Math.SQRT1_2

Value

Math.SQRT1_2; // 0.7071067811865476

Examples

Basic Usage

console.log(Math.SQRT1_2);
// 0.7071067811865476

Verifying the Value

console.log(Math.SQRT1_2 === 1 / Math.sqrt(2));
// true

console.log(Math.SQRT1_2 === Math.sqrt(2) / 2);
// true

Trigonometric Calculations

The value 1/√2 appears in angle calculations:

// cos(45°) = sin(45°) = 1/√2 = √2/2
console.log(Math.cos(Math.PI / 4));
// 0.7071067811865476

console.log(Math.sin(Math.PI / 4));
// 0.7071067811865476

Normalizing a 2D Vector

When normalizing diagonal vectors, you divide by √2:

function normalize2D(x, y) {
  const magnitude = Math.sqrt(x * x + y * y);
  return {
    x: x / magnitude,
    y: y / magnitude
  };
}

// Unit vector along (1, 1) diagonal
const normalized = normalize2D(1, 1);
console.log(normalized.x);
// 0.7071067811865476

console.log(normalized.x === Math.SQRT1_2);
// true

Calculating Diagonal Length

For a unit square, the diagonal length is √2, but each component is 1/√2:

function diagonalComponents(side) {
  return {
    x: side * Math.SQRT1_2,
    y: side * Math.SQRT1_2
  };
}

console.log(diagonalComponents(10));
// { x: 7.071067811865476, y: 7.071067811865476 }

Probability: Normal Distribution

The standard normal distribution uses √2 in its probability density function:

function normalPDF(x) {
  return Math.exp(-0.5 * x * x) / Math.sqrt(2 * Math.PI);
}

console.log(normalPDF(0));
// 0.3989422804014327

Photography: F-Stop Calculations

Aperture ratios relate to √2:

// Each f-stop doubles light, related to √2
function apertureArea(fNumber) {
  return Math.PI / (fNumber * fNumber);
}

console.log(apertureArea(1.414));
// Approximately 1/2 times the area at f/1

Audio: RMS Calculations

Root mean square calculations use √2 for sine waves:

function sineWaveRMS(amplitude) {
  return amplitude / Math.sqrt(2);
}

console.log(sineWaveRMS(1));
// 0.7071067811865476

console.log(sineWaveRMS(1) === Math.SQRT1_2);
// true

Common Patterns

Avoiding Repeated Calculations

// Instead of computing sqrt(1/2) each time
function calculateHalfRoot(value) {
  return value * Math.SQRT1_2;
}

Scale Factor for Unit Circle

// Getting the x or y coordinate at 45 degrees
function unitCircle45() {
  return Math.SQRT1_2;
}

Why Use Math.SQRT1_2?

Using Math.SQRT1_2 instead of hardcoding 0.7071067811865476 provides several advantages:

Precision — The constant is defined to full floating-point precision
Readability — Code clearly expresses the mathematical concept
Standardization — Matches mathematical notation in formulas
Performance — The value is pre-computed by the JavaScript engine
Intent — Using Math.SQRT1_2 signals that 1/√2 is intentionally used