Square Root of 1000

√1000 at a glance

Exact value

10√10

Decimal (10 places)

31.6227766017

Rounded

31.6 · 31.62 · 31.623

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.622777

Prime factorization

2³ × 5³

Cube root

10

How to simplify √1000

Look for the largest perfect square that divides 1000. Here it is 100 (10²), because 1000 = 100 × 10 and 10 has no square factor left:

The prime factorization tells the same story: 1000 = 2³ × 5³. Each pair of equal primes leaves the radical as one factor, so 2 × 5 comes out and 2 × 5 stays inside.

1000 has 3 square factors (4, 25 and 100). Starting with a smaller one still works but takes more rounds: √1000 = 2√250, and √250 can be simplified again. Using 100 straight away finishes in one step.

Check: (10√10)² = 10² × 10 = 100 × 10 = 1000. As a decimal, 10√10 = 10 × 3.1622776602 ≈ 31.6227766017.

Where √1000 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √1000 lies between 31 and 32. 1000 is 39 above 961 and 24 below 1,024, so the root is closer to 32.

• Straight line between 961 and 1,024: 31.6190 (0.01% low)
• Tangent from 31, i.e. 31 + 39 ÷ 62: 31.6290 (0.02% high)
• Tangent from 32, i.e. 32 − 24 ÷ 64: 31.6250 (0.01% high)

For √1000 the tangent at 32 wins, missing by only 0.0022. Tangent estimates shine when the number sits close to a perfect square — here 1000 is just 24 below 1,024.

Finding √1000 with the Babylonian method

Picture a rectangle with an area of 1000 and one side x; the other side must be 1000 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √1000.

Start from the nearest whole number, 32 (32² = 1,024):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √1000 = 31.6227766017 to every decimal shown.

√1000 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √1000 the pattern is [31; 1, 1, 1, 1, 1, 6, 2, 2, 15, 2, 2, 6, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √1000 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 − 1000y² = 1. Its smallest solution in positive whole numbers is x = 39,480,499, y = 1,248,483.

√1000 in geometry and everyday measurements

• 1,000 square feet is 92.9 m². Laid out as a square — a small house footprint or a lot — it is about 31.62 ft (31 ft 7 in) on a side.
• 1000 = 10² + 30² = 18² + 26², so by the Pythagorean theorem √1000 is the diagonal of rectangles measuring 10 × 30 and 18 × 26 — and the distance between the points (0, 0) and (10, 30) on a grid.
• Since √1000 = 10√10, a length of √1000 is exactly 10 copies of the length √10 laid end to end.

1,000 is 10³, so √1,000 = 10 to the power 1.5, or 10√10 ≈ 31.6228 — the point one and a half decades up a logarithmic scale.

Square roots near √1000 and related roots

• The cube root of 1000 is exactly 10 — 1000 is a perfect cube as well (10³).
• Dividing by 100 divides the root by 10: √10 = √1000 ÷ 10 ≈ 3.16227766.

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