Square Root of 432

√432 at a glance

Exact value

12√3

Decimal (10 places)

20.7846096908

Rounded

20.8 · 20.78 · 20.785

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.784610

Prime factorization

2⁴ × 3³

Cube root

7.559526

How to simplify √432

Look for the largest perfect square that divides 432. Here it is 144 (12²), because 432 = 144 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 432 = 2⁴ × 3³. Each pair of equal primes leaves the radical as one factor, so 2² × 3 comes out and 3 stays inside.

432 has 5 square factors (4, 9, 16, 36 and 144). Starting with a smaller one still works but takes more rounds: √432 = 2√108, and √108 can be simplified again. Using 144 straight away finishes in one step.

Check: (12√3)² = 12² × 3 = 144 × 3 = 432. As a decimal, 12√3 = 12 × 1.7320508076 ≈ 20.7846096908.

Where √432 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √432 lies between 20 and 21. 432 is 32 above 400 and 9 below 441, so the root is closer to 21.

• Straight line between 400 and 441: 20.7805 (0.02% low)
• Tangent from 20, i.e. 20 + 32 ÷ 40: 20.8000 (0.07% high)
• Tangent from 21, i.e. 21 − 9 ÷ 42: 20.7857 (0.01% high)

For √432 the tangent at 21 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 432 is just 9 below 441.

Finding √432 with the Babylonian method

Picture a rectangle with an area of 432 and one side x; the other side must be 432 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √432.

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 √432 = 20.7846096908 to every decimal shown.

√432 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √432 the pattern is [20; 1, 3, 1, 1, 1, 3, 1, 40] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √432 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 − 432y² = 1. Its smallest solution in positive whole numbers is x = 1,351, y = 65.

√432 in geometry and everyday measurements

• A square garage floor of 432 square feet measures about 20.78 ft (20 ft 9 in) per side, and its corner-to-corner diagonal is √864 ≈ 29.4 ft.
• 432 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √432 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 4 × 20 box, because 4² + 4² + 20² = 432.
• Since √432 = 12√3, a length of √432 is exactly 12 copies of the length √3 laid end to end.

Square roots near √432 and related roots

• The cube root of 432 is about 7.559526.
• Because 432 = 4 × 108, the root is twice √108: 2 × 10.392305 ≈ 20.78461.

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