Square Root of 436

√436 at a glance

Exact value

2√109

Decimal (10 places)

20.8806130178

Rounded

20.9 · 20.88 · 20.881

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.880613

Prime factorization

2² × 109

Cube root

7.582787

How to simplify √436

Look for the largest perfect square that divides 436. Here it is 4 (2²), because 436 = 4 × 109 and 109 has no square factor left:

The prime factorization tells the same story: 436 = 2² × 109. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 109 stays inside.

Check: (2√109)² = 2² × 109 = 4 × 109 = 436. As a decimal, 2√109 = 2 × 10.4403065089 ≈ 20.8806130178.

Where √436 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √436 lies between 20 and 21. 436 is 36 above 400 and 5 below 441, so the root is closer to 21.

• Straight line between 400 and 441: 20.8780 (0.01% low)
• Tangent from 20, i.e. 20 + 36 ÷ 40: 20.9000 (0.09% high)
• Tangent from 21, i.e. 21 − 5 ÷ 42: 20.8810 (0% high)

For √436 the tangent at 21 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 436 is just 5 below 441.

Finding √436 with the Babylonian method

Picture a rectangle with an area of 436 and one side x; the other side must be 436 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √436.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √436 = 20.8806130178 to every decimal shown.

√436 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √436 the pattern is [20; 1, 7, 2, 1, 1, 1, 13, 3, 2, 2, 5, 1, …] with the block of 30 terms after the semicolon repeating forever (only the first 12 of the 30 are shown). A pattern that never ends is one more proof that √436 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 − 436y² = 1. Its smallest solution in positive whole numbers is x = 158,070,671,986,249, y = 7,570,212,227,550 — 15 digits for x, even though 436 is small, which is what makes Pell’s equation famous.

√436 in geometry and everyday measurements

• A square garage floor of 436 square feet measures about 20.88 ft (20 ft 11 in) per side, and its corner-to-corner diagonal is √872 ≈ 29.5 ft.
• 436 = 6² + 20², so by the Pythagorean theorem √436 is the diagonal of a 6 × 20 rectangle — and the distance between the points (0, 0) and (6, 20) on a grid.
• Since √436 = 2√109, a length of √436 is exactly 2 copies of the length √109 laid end to end.

Square roots near √436 and related roots

• The cube root of 436 is about 7.582787.
• Because 436 = 4 × 109, the root is twice √109: 2 × 10.440307 ≈ 20.880613.

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