Ever wondered how to work out square roots without a...
Maths206 views·Updated 24 Aug 2026·2 pagesEstimating Square Roots Without a Calculator
Scarlett Edwardes@scarlett_1917
1
of 2
Estimating Square Roots - The Basics
When you need to estimate square roots without a calculator, you're essentially playing a clever guessing game. The key is finding the perfect square that's closest to your number.
Let's say you need to find √83. You know that 9² = 81, which is only 2 away from 83. Since 83 is just slightly bigger than 81, √83 must be just over 9 - roughly 9.1.
The method works because you're thinking about where your number sits between two perfect squares. Since the next perfect square after 81 is 100 (which is 10²), and 83 is much closer to 81, your estimate will be just slightly above 9.
Quick Tip: Always identify the perfect squares on either side of your number first - this gives you the "boundaries" for your estimate.
2
of 2
Step-by-Step Examples
Example 1: √26 First, spot that 25 (which is 5²) is the closest perfect square, just 1 away from 26. The next perfect square is 36 (6²). Since 26 is very close to 25 and far from 36, √26 ≈ 5.1.
Example 2: √110 Here, 100 (10²) is closest, being 10 away from 110. The next perfect square is 121 (11²). Notice that 110 sits roughly halfway between 100 and 121, so √110 ≈ 10.5.
The pattern is simple: if your number is close to the smaller perfect square, add a small decimal. If it's roughly in the middle, add around 0.5.
Remember: You're not aiming for perfection here - just a reasonable estimate that gets you in the right ballpark!
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Maths206 views·Updated 24 Aug 2026·2 pagesEstimating Square Roots Without a Calculator
Scarlett Edwardes@scarlett_1917
Ever wondered how to work out square roots without a calculator when the number isn't a perfect square? It's actually quite straightforward once you know the trick of finding the closest perfect squares and making smart estimates.
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of 2
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Estimating Square Roots - The Basics
When you need to estimate square roots without a calculator, you're essentially playing a clever guessing game. The key is finding the perfect square that's closest to your number.
Let's say you need to find √83. You know that 9² = 81, which is only 2 away from 83. Since 83 is just slightly bigger than 81, √83 must be just over 9 - roughly 9.1.
The method works because you're thinking about where your number sits between two perfect squares. Since the next perfect square after 81 is 100 (which is 10²), and 83 is much closer to 81, your estimate will be just slightly above 9.
Quick Tip: Always identify the perfect squares on either side of your number first - this gives you the "boundaries" for your estimate.
2
of 2Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
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Step-by-Step Examples
Example 1: √26 First, spot that 25 (which is 5²) is the closest perfect square, just 1 away from 26. The next perfect square is 36 (6²). Since 26 is very close to 25 and far from 36, √26 ≈ 5.1.
Example 2: √110 Here, 100 (10²) is closest, being 10 away from 110. The next perfect square is 121 (11²). Notice that 110 sits roughly halfway between 100 and 121, so √110 ≈ 10.5.
The pattern is simple: if your number is close to the smaller perfect square, add a small decimal. If it's roughly in the middle, add around 0.5.
Remember: You're not aiming for perfection here - just a reasonable estimate that gets you in the right ballpark!
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
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