Square Root of 972

√972 at a glance

Exact value

18√3

Decimal (10 places)

31.1769145362

Rounded

31.2 · 31.18 · 31.177

Perfect square?

No — between 31² and 32²

Rational?

Irrational

Both square roots

±31.176915

Prime factorization

2² × 3⁵

Cube root

9.905782

How to simplify √972

Look for the largest perfect square that divides 972. Here it is 324 (18²), because 972 = 324 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 972 = 2² × 3⁵. Each pair of equal primes leaves the radical as one factor, so 2 × 3² comes out and 3 stays inside.

972 has 5 square factors (4, 9, 36, 81 and 324). Starting with a smaller one still works but takes more rounds: √972 = 2√243, and √243 can be simplified again. Using 324 straight away finishes in one step.

Check: (18√3)² = 18² × 3 = 324 × 3 = 972. As a decimal, 18√3 = 18 × 1.7320508076 ≈ 31.1769145362.

Where √972 sits between perfect squares

961 = 31² and 1,024 = 32² are the nearest perfect squares, so √972 lies between 31 and 32. 972 is 11 above 961 and 52 below 1,024, so the root is closer to 31.

• Straight line between 961 and 1,024: 31.1746 (0.01% low)
• Tangent from 31, i.e. 31 + 11 ÷ 62: 31.1774 (0% high)
• Tangent from 32, i.e. 32 − 52 ÷ 64: 31.1875 (0.03% high)

For √972 the tangent at 31 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 972 is just 11 above 961.

Finding √972 with the Babylonian method

Picture a rectangle with an area of 972 and one side x; the other side must be 972 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √972.

Start from the nearest whole number, 31 (31² = 961):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √972 = 31.1769145362 to every decimal shown.

√972 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √972 the pattern is [31; 5, 1, 1, 1, 7, 6, 1, 3, 1, 14, 1, 3, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √972 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 − 972y² = 1. Its smallest solution in positive whole numbers is x = 9,863,382,151, y = 316,368,130.

√972 in geometry and everyday measurements

• 972 square feet is 90.3 m². Laid out as a square — a small house footprint or a lot — it is about 31.18 ft (31 ft 2 in) on a side.
• 972 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 √972 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 22 × 22 box, because 2² + 22² + 22² = 972.
• Since √972 = 18√3, a length of √972 is exactly 18 copies of the length √3 laid end to end.

Square roots near √972 and related roots

• The cube root of 972 is about 9.905782.
• Because 972 = 4 × 243, the root is twice √243: 2 × 15.588457 ≈ 31.176915.

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