Square Root of 452

√452 at a glance

Exact value

2√113

Decimal (10 places)

21.2602916255

Rounded

21.3 · 21.26 · 21.260

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.260292

Prime factorization

2² × 113

Cube root

7.674430

How to simplify √452

Look for the largest perfect square that divides 452. Here it is 4 (2²), because 452 = 4 × 113 and 113 has no square factor left:

The prime factorization tells the same story: 452 = 2² × 113. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 113 stays inside.

Check: (2√113)² = 2² × 113 = 4 × 113 = 452. As a decimal, 2√113 = 2 × 10.6301458127 ≈ 21.2602916255.

Where √452 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √452 lies between 21 and 22. 452 is 11 above 441 and 32 below 484, so the root is closer to 21.

• Straight line between 441 and 484: 21.2558 (0.02% low)
• Tangent from 21, i.e. 21 + 11 ÷ 42: 21.2619 (0.01% high)
• Tangent from 22, i.e. 22 − 32 ÷ 44: 21.2727 (0.06% high)

For √452 the tangent at 21 wins, missing by only 0.0016. Tangent estimates shine when the number sits close to a perfect square — here 452 is just 11 above 441.

Finding √452 with the Babylonian method

Picture a rectangle with an area of 452 and one side x; the other side must be 452 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √452.

Start from the nearest whole number, 21 (21² = 441):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √452 = 21.2602916255 to every decimal shown.

√452 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √452 the pattern is [21; 3, 1, 5, 3, 10, 3, 5, 1, 3, 42] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √452 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 452y² = 1. Its smallest solution in positive whole numbers is x = 1,204,353, y = 56,648.

√452 in geometry and everyday measurements

• A square garage floor of 452 square feet measures about 21.26 ft (21 ft 3 in) per side, and its corner-to-corner diagonal is √904 ≈ 30.1 ft.
• 452 = 14² + 16², so by the Pythagorean theorem √452 is the diagonal of a 14 × 16 rectangle — and the distance between the points (0, 0) and (14, 16) on a grid.
• Since √452 = 2√113, a length of √452 is exactly 2 copies of the length √113 laid end to end.

Square roots near √452 and related roots

• The cube root of 452 is about 7.674430.
• Because 452 = 4 × 113, the root is twice √113: 2 × 10.630146 ≈ 21.260292.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.