Square Root of 85 — Value, Method & How to Find

The Answer At A Glance

Result: $\sqrt{85} \approx 9.21954445729$

Notation: Decimal approximation; exact form is $\sqrt{85}$.

Method shown: Long division method (manual), with cross-checks using Newton's method and linear interpolation.

Approximate value: $9.2195$ (4 d.p.)

Exact form: $\sqrt{85}$ — cannot be simplified, since $85 = 5 \times 17$ has no square factor.

Quick Reference Table — Square Roots Near 85

Between $\sqrt{81} = 9$ and $\sqrt{100} = 10$, $\sqrt{85}$ sits closer to the lower end at $9.22$. Only $81$ and $100$ in this band have integer square roots.

What "Square Root of 85" Means

The square root of a non-negative number $n$ is the value $x$ such that $x^2 = n$. For $\sqrt{85}$, that means the positive number $x$ for which $x \cdot x = 85$.

Because $9^2 = 81$ and $10^2 = 100$, $\sqrt{85}$ falls between $9$ and $10$ — closer to $9$ than to $10$ because $85$ is closer to $81$.

Is √85 Rational or Irrational?

$\sqrt{85}$ is irrational. Reason: $85 = 5 \times 17$. Neither $5$ nor $17$ is a perfect square, and a number is a perfect square if and only if every prime in its factorisation appears to an even power. Since both $5$ and $17$ appear to the first power, $85$ is not a perfect square, and its square root cannot be written as a fraction $p/q$.

The decimal $9.21954445729\dots$ neither terminates nor repeats — the definition of irrational.

How To Find √85 — Three Methods

Method 1 — Long division (digit by digit)

Pair the digits of $85$: $85.000000$.

Step 1. Largest integer with square $\leq 85$ is $9$ ($9^2 = 81$). Subtract: $85 - 81 = 4$. Bring down $00$: $400$.

Step 2. Double $9$: $18$. Find $d$ with $(180 + d) \cdot d \leq 400$. $d = 2$ gives $182 \cdot 2 = 364$. Subtract: $400 - 364 = 36$. Bring down $00$: $3600$.

Step 3. Double $9.2$: $184$. Find $d$ with $(1840 + d) \cdot d \leq 3600$. $d = 1$ gives $1841$. Subtract: $3600 - 1841 = 1759$. Bring down $00$: $175{,}900$.

Step 4. Double $9.21$: $1842$. $d = 9$ gives $18429 \cdot 9 = 165{,}861$. Subtract.

After four steps: $\sqrt{85} \approx 9.219$. Continuing gives $9.2195$.

Final answer: $\sqrt{85} \approx 9.2195$.

Method 2 — Newton's iteration (fastest)

$$x_{k+1} = \frac{1}{2}\left(x_k + \frac{n}{x_k}\right)$$

Start $x_0 = 9$.

• $x_1 = \frac{1}{2}(9 + 85/9) = \frac{1}{2}(9 + 9.444) = 9.2222$

• $x_2 = \frac{1}{2}(9.2222 + 85/9.2222) = \frac{1}{2}(9.2222 + 9.2169) = 9.2195$

Two iterations to four-decimal precision. Newton doubles the correct digits each step.

Method 3 — Linear interpolation (mental estimate)

$$\sqrt{85} \approx 9 + \frac{85 - 81}{100 - 81} = 9 + \frac{4}{19} \approx 9.21$$

Quick enough for a sanity check.

Where √85 Shows Up

$\sqrt{85}$ appears as the diagonal of a $2 \times 9$ rectangle — Pythagoras gives $\sqrt{4 + 81} = \sqrt{85}$. It also turns up in the distance between $(0, 0)$ and $(6, 7)$: $\sqrt{36 + 49} = \sqrt{85}$. In surveying and navigation, irrational roots of the form $\sqrt{a^2 + b^2}$ are routine — $\sqrt{85}$ is one of the small ones that arise on integer-grid coordinates.

Three Slips That Cost Marks on √85

Mistake 1: Trying to simplify when no square factor exists.

Where it slips in: Students assume every non-perfect-square integer simplifies to $a\sqrt{b}$.

Don't do this: $\sqrt{85} = \sqrt{5 \cdot 17}$ → "simplifies to" something cleaner.

The correct way: Check whether the radicand has a perfect-square factor other than $1$. $85 = 5 \times 17$ — no square factor. $\sqrt{85}$ is already in simplest radical form.

In Bhanzu's Grade 8 cohorts, the "must simplify" reflex shows up on roughly four out of ten first attempts at $\sqrt{85}$ — students try to force a factorisation that does not exist.

Mistake 2: Reporting a truncated decimal as exact.

Don't do this: $\sqrt{85} = 9.2195$.

The correct way: $\sqrt{85} \approx 9.2195$ (4 d.p.). The exact value is $\sqrt{85}$ — irrational decimals never terminate.

Mistake 3: Confusing $\sqrt{85}$ with $\pm\sqrt{85}$.

Where it slips in: Solving $x^2 = 85$.

Don't do this: $x^2 = 85 \implies x = \sqrt{85}$ (only).

The correct way: $x^2 = 85 \implies x = \pm\sqrt{85}$. Both square to $85$. The notation $\sqrt{85}$ alone refers to the positive root. A Bhanzu trainer at the McKinney TX center keeps the $\pm$ habit on the side board for the first month of root work — the dropped negative is the single most common slip.

Conclusion

• The square root of 85 is approximately $9.2195$ — irrational, non-terminating, non-repeating.

• $85 = 5 \times 17$ has no perfect-square factor, so $\sqrt{85}$ cannot be simplified.

• Three methods compute it: long division, Newton's iteration (fastest), and linear interpolation (quickest mental estimate).

• Always use $\approx$ for irrational decimal approximations.

• $\sqrt{85}$ shows up as the diagonal of a $2 \times 9$ rectangle and in standard distance-formula calculations.

A practical next step

1. Find $\sqrt{84}$ to two decimal places using Newton's method starting from $x_0 = 9$.

2. Is $\sqrt{144}$ rational or irrational? Justify by prime factorisation.

3. The diagonal of a rectangle is $\sqrt{85}$ cm and one side is $7$ cm. Find the other side.

Want a Bhanzu trainer to walk you through more square-root problems? Book a free demo class — online globally.