Square Root of 652

√652 at a glance

Exact value

2√163

Decimal (10 places)

25.5342906696

Rounded

25.5 · 25.53 · 25.534

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.534291

Prime factorization

2² × 163

Cube root

8.671266

How to simplify √652

Look for the largest perfect square that divides 652. Here it is 4 (2²), because 652 = 4 × 163 and 163 has no square factor left:

The prime factorization tells the same story: 652 = 2² × 163. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 163 stays inside.

Check: (2√163)² = 2² × 163 = 4 × 163 = 652. As a decimal, 2√163 = 2 × 12.7671453348 ≈ 25.5342906696.

Where √652 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √652 lies between 25 and 26. 652 is 27 above 625 and 24 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.5294 (0.02% low)
• Tangent from 25, i.e. 25 + 27 ÷ 50: 25.5400 (0.02% high)
• Tangent from 26, i.e. 26 − 24 ÷ 52: 25.5385 (0.02% high)

For √652 the tangent at 26 wins, missing by only 0.0042. Tangent estimates shine when the number sits close to a perfect square — here 652 is just 24 below 676.

Finding √652 with the Babylonian method

Picture a rectangle with an area of 652 and one side x; the other side must be 652 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √652.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √652 = 25.5342906696 to every decimal shown.

√652 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √652 the pattern is [25; 1, 1, 6, 1, 3, 1, 3, 2, 5, 1, 16, 5, …] with the block of 36 terms after the semicolon repeating forever (only the first 12 of the 36 are shown). A pattern that never ends is one more proof that √652 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 − 652y² = 1. Its smallest solution in positive whole numbers is x = 8,212,499,464,321,351, y = 321,626,301,297,510 — 16 digits for x, even though 652 is small, which is what makes Pell’s equation famous.

√652 in geometry and everyday measurements

• 652 square feet is 60.6 m². Laid out as a square — a small house footprint or a lot — it is about 25.53 ft (25 ft 6 in) on a side.
• 652 is not a sum of two whole-number squares — the prime factor 163 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √652 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 18 × 18 box, because 2² + 18² + 18² = 652.
• Since √652 = 2√163, a length of √652 is exactly 2 copies of the length √163 laid end to end.

Square roots near √652 and related roots

• The cube root of 652 is about 8.671266.
• Because 652 = 4 × 163, the root is twice √163: 2 × 12.767145 ≈ 25.534291.

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