How to Approximate Square Roots — Formulae, Tricks & Examples

Need a square root estimate without a calculator? This guide explains fast, accurate methods for approximating square roots—including formulas, common tricks, and hands-on examples. Perfect for students, educators, math fans and exam prep!

The Square Root Function

\[ \sqrt{x} \]

Our goal: estimate values of \( \sqrt{x} \) when \( x \) is not a perfect square.

Common Approximations and Square Root Benchmarks

\( x \)	\( \sqrt{x} \) (Exact)	Common Approximations
2	\( 1.414 \ldots \)	1.41
3	\( 1.732 \ldots \)	1.73
5	\( 2.236 \ldots \)	2.24
7	\( 2.645 \ldots \)	2.65
10	\( 3.162 \ldots \)	3.16
11	\( 3.317 \ldots \)	3.32
12	\( 3.464 \ldots \)	3.46
15	\( 3.873 \ldots \)	3.87
20	\( 4.472 \ldots \)	4.47
50	\( 7.071 \ldots \)	7.07
100	10	10

Memorize root values for key numbers—these help estimate other roots quickly.

Key Methods for Approximating Square Roots

1. The Average Method (Babylonian/Heron's Method)

If you want \( \sqrt{x} \):
Pick an initial guess \( g_0 > 0 \) (start close to the root).
Iterate:

\[ g_{n+1} = \frac{1}{2} \left(g_n + \frac{x}{g_n}\right) \]

Repeat until results stabilize (usually 2–4 rounds).

Example: Estimate \( \sqrt{10} \) — let \( g_0 = 3 \).

First iteration: \( g_1 = \frac{1}{2} (3 + \frac{10}{3}) = \frac{1}{2} (3 + 3.\overline{3}) = 3.1666 \ldots \)

Second iteration: \( g_2 = \frac{1}{2} (3.1666 + \frac{10}{3.1666}) \approx 3.1623 \)

So \( \sqrt{10} \approx 3.16 \) after only two steps.

2. Linear Approximation Formula (Newton’s Estimate)

Linearization (for \( x \) near \( a \)):

\[ \sqrt{x} \approx \sqrt{a} + \frac{x-a}{2\sqrt{a}} \]

Where \( a \) is a perfect square near \( x \).

Example: Approximate \( \sqrt{11} \) (take \( a = 9 \), so \( \sqrt{9} = 3 \))

\( \sqrt{11} \approx 3 + \frac{11-9}{2 \times 3} = 3 + \frac{2}{6} = 3.333 \)

(Actual value ≈ 3.317 – this method is good for quick mental math!)

3. Side-by-Side Table of Square Root Approximations

\( x \)	\( \sqrt{x} \) Estimate (Babylonian, 2 rounds)	\( \sqrt{x} \) Estimate (Linear approx.)	Actual	Error (%)
2	1.414	1.5	1.4142	+6.1
3	1.732	1.75	1.7321	+1.0
5	2.236	2.25	2.2361	+0.6
7	2.646	2.67	2.6458	+0.9
10	3.162	3.167	3.1623	+0.1
11	3.317	3.333	3.3166	+0.5
20	4.472	4.5	4.4721	+0.6

Frequently Asked Questions

Q: Does the Babylonian method always work?
A: Yes! It converges quickly for any positive number and is widely used in calculators for square root computations.

Q: Do these formulas work for cube roots?
A: They can be generalized, but the iteration changes: \[ g_{n+1} = \frac{2g_n + \frac{x}{g_n^2}}{3} \] (for \( \sqrt[3]{x} \))

Q: How do I estimate square roots of decimals?
A: Use the same formulas—just keep track of your decimal places. For example,
\( \sqrt{0.2} \approx 0.45 \) (since \( 0.2 \) is close to \( 0.25 \) and \( \sqrt{0.25} = 0.5 \))

Try using the Babylonian method or linearization to estimate roots during exams or mental math! For even faster results, memorize root values for 2, 3, 5, 10, and 20.

Ready to Master Square Roots?

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