Square Root of 87

√87 at a glance

Exact value

√87

Decimal (10 places)

9.3273790531

Rounded

9.3 · 9.33 · 9.327

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.327379

Prime factorization

3 × 29

Cube root

4.431048

How to simplify √87

The prime factorization of 87 is 3 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √87 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 87, 3 and 29 appear an odd number of times, so √87 is irrational and 9.3273790531 is a rounded value.

Where √87 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √87 lies between 9 and 10. 87 is 6 above 81 and 13 below 100, so the root is closer to 9.

• Straight line between 81 and 100: 9.3158 (0.12% low)
• Tangent from 9, i.e. 9 + 6 ÷ 18: 9.3333 (0.06% high)
• Tangent from 10, i.e. 10 − 13 ÷ 20: 9.3500 (0.24% high)

For √87 the tangent at 9 wins, missing by only 0.006. Tangent estimates shine when the number sits close to a perfect square — here 87 is just 6 above 81.

Finding √87 with the Babylonian method

If a guess is too big, 87 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√87) in one step.

Start from the nearest whole number, 9 (9² = 81):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √87 = 9.3273790531 to every decimal shown.

√87 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √87 the pattern is [9; 3, 18] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √87 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 − 87y² = 1. Its smallest solution in positive whole numbers is x = 28, y = 3.

√87 in geometry and everyday measurements

• A square room or garden bed covering 87 square feet measures about 9.33 ft (9 ft 4 in) along each wall.
• 87 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 √87 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √87 as its space diagonal.

Square roots near √87 and related roots

• The cube root of 87 is about 4.431048.
• Four times the radicand doubles the root: √348 = 2 × √87 ≈ 18.654758.

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